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Precalculus

4.5 Logarithmic Properties

Precalculus4.5 Logarithmic Properties
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Learning Objectives

In this section, you will:

  • Use the product rule for logarithms.
  • Use the quotient rule for logarithms.
  • Use the power rule for logarithms.
  • Expand logarithmic expressions.
  • Condense logarithmic expressions.
  • Use the change-of-base formula for logarithms.
Testing of the pH of hydrochloric acid.
Figure 1 The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan)

In chemistry, pH is used as a measure of the acidity or alkalinity of a substance. The pH scale runs from 0 to 14. Substances with a pH less than 7 are considered acidic, and substances with a pH greater than 7 are said to be alkaline. Our bodies, for instance, must maintain a pH close to 7.35 in order for enzymes to work properly. To get a feel for what is acidic and what is alkaline, consider the following pH levels of some common substances:

  • Battery acid: 0.8
  • Stomach acid: 2.7
  • Orange juice: 3.3
  • Pure water: 7 (at 25° C)
  • Human blood: 7.35
  • Fresh coconut: 7.8
  • Sodium hydroxide (lye): 14

To determine whether a solution is acidic or alkaline, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where H+ H+ is the concentration of hydrogen ion in the solution

pH=log([ H + ]) =log( 1 [ H + ] ) pH=log([ H + ]) =log( 1 [ H + ] )

The equivalence of log( [ H + ] ) log( [ H + ] ) and log( 1 [ H + ] ) log( 1 [ H + ] ) is one of the logarithm properties we will examine in this section.

Using the Product Rule for Logarithms

Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove.

log b 1=0 log b b=1 log b 1=0 log b b=1

For example, log 5 1=0 log 5 1=0 since 5 0 =1. 5 0 =1. And log 5 5=1 log 5 5=1 since 5 1 =5. 5 1 =5.

Next, we have the inverse property.

log b ( b x )=x    b log b x =x,x>0 log b ( b x )=x    b log b x =x,x>0

For example, to evaluate log( 100 ), log( 100 ), we can rewrite the logarithm as log 10 ( 10 2 ), log 10 ( 10 2 ), and then apply the inverse property log b ( b x )=x log b ( b x )=x to get log 10 ( 10 2 )=2. log 10 ( 10 2 )=2.

To evaluate e ln( 7 ) , e ln( 7 ) , we can rewrite the logarithm as e log e 7 , e log e 7 , and then apply the inverse property b log b x =x b log b x =x to get e log e 7 =7. e log e 7 =7.

Finally, we have the one-to-one property.

log b M= log b Nif and only ifM=N log b M= log b Nif and only ifM=N

We can use the one-to-one property to solve the equation log 3 ( 3x )= log 3 ( 2x+5 ) log 3 ( 3x )= log 3 ( 2x+5 ) for x. x. Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for x: x:

3x=2x+5 Set the arguments equal. x=5 Subtract 2x. 3x=2x+5 Set the arguments equal. x=5 Subtract 2x.

But what about the equation log 3 ( 3x )+ log 3 ( 2x+5 )=2? log 3 ( 3x )+ log 3 ( 2x+5 )=2? The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining terms on the left side of the equation.

Recall that we use the product rule of exponents to combine the product of exponents by adding: x a x b = x a+b . x a x b = x a+b . We have a similar property for logarithms, called the product rule for logarithms, which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.

Given any real number x x and positive real numbers M,N, M,N, and b, b, where b1, b1, we will show

log b ( MN )= log b ( M )+ log b ( N ). log b ( MN )= log b ( M )+ log b ( N ).

Let m= log b M m= log b M and n= log b N. n= log b N. In exponential form, these equations are b m =M b m =M and b n =N. b n =N. It follows that

log b ( MN ) = log b ( b m b n ) Substitute for Mand N. = log b ( b m+n ) Apply the product rule for exponents. =m+n Apply the inverse property of logs. = log b ( M )+ log b ( N ) Substitute for mand n. log b ( MN ) = log b ( b m b n ) Substitute for Mand N. = log b ( b m+n ) Apply the product rule for exponents. =m+n Apply the inverse property of logs. = log b ( M )+ log b ( N ) Substitute for mand n.

Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider log b (wxyz). log b (wxyz). Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:

log b (wxyz)= log b w+ log b x+ log b y+ log b z log b (wxyz)= log b w+ log b x+ log b y+ log b z

The Product Rule for Logarithms

The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.

log b (MN)= log b ( M )+ log b ( N )for b>0 log b (MN)= log b ( M )+ log b ( N )for b>0

How To

Given the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.

  1. Factor the argument completely, expressing each whole number factor as a product of primes.
  2. Write the equivalent expression by summing the logarithms of each factor.

Example 1

Using the Product Rule for Logarithms

Expand log 3 ( 30x( 3x+4 ) ). log 3 ( 30x( 3x+4 ) ).

Try It #1

Expand log b (8k). log b (8k).

Using the Quotient Rule for Logarithms

For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to combine the quotient of exponents by subtracting: x a x b = x ab . x a x b = x ab . The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.

Given any real number x x and positive real numbers M, M, N, N, and b, b, where b1, b1, we will show

log b ( M N )= log b ( M ) log b ( N ). log b ( M N )= log b ( M ) log b ( N ).

Let m= log b M m= log b M and n= log b N. n= log b N. In exponential form, these equations are b m =M b m =M and b n =N. b n =N. It follows that

log b ( M N ) = log b ( b m b n ) Substitute for Mand N. = log b ( b mn ) Apply the quotient rule for exponents. =mn Apply the inverse property of logs. = log b ( M ) log b ( N ) Substitute for mand n. log b ( M N ) = log b ( b m b n ) Substitute for Mand N. = log b ( b mn ) Apply the quotient rule for exponents. =mn Apply the inverse property of logs. = log b ( M ) log b ( N ) Substitute for mand n.

For example, to expand log( 2 x 2 +6x 3x+9 ), log( 2 x 2 +6x 3x+9 ), we must first express the quotient in lowest terms. Factoring and canceling we get,

log( 2 x 2 +6x 3x+9 )=log( 2x(x+3) 3(x+3) ) Factor the numerator and denominator.                       =log( 2x 3 ) Cancel the common factors. log( 2 x 2 +6x 3x+9 )=log( 2x(x+3) 3(x+3) ) Factor the numerator and denominator.                       =log( 2x 3 ) Cancel the common factors.

Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule.

log( 2x 3 )=log(2x)log(3)             =log(2)+log(x)log(3) log( 2x 3 )=log(2x)log(3)             =log(2)+log(x)log(3)

The Quotient Rule for Logarithms

The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.

log b ( M N )= log b M log b N log b ( M N )= log b M log b N

How To

Given the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms.

  1. Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.
  2. Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.
  3. Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.

Example 2

Using the Quotient Rule for Logarithms

Expand log 2 ( 15x(x1) (3x+4)(2x) ). log 2 ( 15x(x1) (3x+4)(2x) ).

Analysis

There are exceptions to consider in this and later examples. First, because denominators must never be zero, this expression is not defined for x= 4 3 x= 4 3 and x=2. x=2. Also, since the argument of a logarithm must be positive, we note as we observe the expanded logarithm, that x>0, x>0, x>1, x>1, x> 4 3 , x> 4 3 , and x<2. x<2. Combining these conditions is beyond the scope of this section, and we will not consider them here or in subsequent exercises.

Try It #2

Expand log 3 ( 7 x 2 +21x 7x( x1 )( x2 ) ). log 3 ( 7 x 2 +21x 7x( x1 )( x2 ) ).

Using the Power Rule for Logarithms

We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as x 2 ? x 2 ? One method is as follows:

log b ( x 2 ) = log b ( xx ) = log b x+ log b x =2 log b x log b ( x 2 ) = log b ( xx ) = log b x+ log b x =2 log b x

Notice that we used the product rule for logarithms to find a solution for the example above. By doing so, we have derived the power rule for logarithms, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,

100= 10 2 3 = 3 1 2 1 e = e 1 100= 10 2 3 = 3 1 2 1 e = e 1

The Power Rule for Logarithms

The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.

log b ( M n )=n log b M log b ( M n )=n log b M

How To

Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.

  1. Express the argument as a power, if needed.
  2. Write the equivalent expression by multiplying the exponent times the logarithm of the base.

Example 3

Expanding a Logarithm with Powers

Expand log 2 x 5 . log 2 x 5 .

Try It #3

Expand ln x 2 . ln x 2 .

Example 4

Rewriting an Expression as a Power before Using the Power Rule

Expand log 3 ( 25 ) log 3 ( 25 ) using the power rule for logs.

Try It #4

Expand ln( 1 x 2 ). ln( 1 x 2 ).

Example 5

Using the Power Rule in Reverse

Rewrite 4ln(x) 4ln(x) using the power rule for logs to a single logarithm with a leading coefficient of 1.

Try It #5

Rewrite 2 log 3 4 2 log 3 4 using the power rule for logs to a single logarithm with a leading coefficient of 1.

Expanding Logarithmic Expressions

Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.” Sometimes we apply more than one rule in order to simplify an expression. For example:

log b ( 6x y ) = log b ( 6x ) log b y = log b 6+ log b x log b y log b ( 6x y ) = log b ( 6x ) log b y = log b 6+ log b x log b y

We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power:

log b ( A C ) = log b ( A C 1 ) = log b ( A )+ log b ( C 1 ) = log b A+(1) log b C = log b A log b C log b ( A C ) = log b ( A C 1 ) = log b ( A )+ log b ( C 1 ) = log b A+(1) log b C = log b A log b C

We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.

With practice, we can look at a logarithmic expression and expand it mentally, writing the final answer. Remember, however, that we can only do this with products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm.

Example 6

Expanding Logarithms Using Product, Quotient, and Power Rules

Rewrite ln( x 4 y 7 ) ln( x 4 y 7 ) as a sum or difference of logs.

Try It #6

Expand log( x 2 y 3 z 4 ). log( x 2 y 3 z 4 ).

Example 7

Using the Power Rule for Logarithms to Simplify the Logarithm of a Radical Expression

Expand log( x ). log( x ).

Try It #7

Expand ln( x 2 3 ). ln( x 2 3 ).

Q&A

Can we expand ln( x 2 + y 2 )? ln( x 2 + y 2 )?

No. There is no way to expand the logarithm of a sum or difference inside the argument of the logarithm.

Example 8

Expanding Complex Logarithmic Expressions

Expand log 6 ( 64 x 3 ( 4x+1 ) ( 2x1 ) ). log 6 ( 64 x 3 ( 4x+1 ) ( 2x1 ) ).

Try It #8

Expand ln( (x1) (2x+1) 2 ( x 2 9) ). ln( (x1) (2x+1) 2 ( x 2 9) ).

Condensing Logarithmic Expressions

We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing.

How To

Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm.

  1. Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power.
  2. Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.
  3. Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.

Example 9

Using the Product and Quotient Rules to Combine Logarithms

Write log 3 ( 5 )+ log 3 ( 8 ) log 3 ( 2 ) log 3 ( 5 )+ log 3 ( 8 ) log 3 ( 2 ) as a single logarithm.

Try It #9

Condense log3log4+log5log6. log3log4+log5log6.

Example 10

Condensing Complex Logarithmic Expressions

Condense log 2 ( x 2 )+ 1 2 log 2 ( x1 )3 log 2 ( ( x+3 ) 2 ). log 2 ( x 2 )+ 1 2 log 2 ( x1 )3 log 2 ( ( x+3 ) 2 ).

Try It #10

Rewrite log( 5 )+0.5log( x )log( 7x1 )+3log( x1 ) log( 5 )+0.5log( x )log( 7x1 )+3log( x1 ) as a single logarithm.

Example 11

Rewriting as a Single Logarithm

Rewrite 2logx4log(x+5)+ 1 x log( 3x+5 ) 2logx4log(x+5)+ 1 x log( 3x+5 ) as a single logarithm.

Try It #11

Condense 4( 3log( x )+log( x+5 )log( 2x+3 ) ). 4( 3log( x )+log( x+5 )log( 2x+3 ) ).

Example 12

Applying of the Laws of Logs

Recall that, in chemistry, pH=log[ H + ]. pH=log[ H + ]. If the concentration of hydrogen ions in a liquid is doubled, what is the effect on pH?

Try It #12

How does the pH change when the concentration of positive hydrogen ions is decreased by half?

Using the Change-of-Base Formula for Logarithms

Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10 or e, e, we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.

To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms.

Given any positive real numbers M,b, M,b, and n, n, where n1  n1  and b1, b1, we show

log b M= log n M log n b log b M= log n M log n b

Let y= log b M. y= log b M. By taking the log base n n of both sides of the equation, we arrive at an exponential form, namely b y =M. b y =M. It follows that

log n ( b y ) = log n M Apply the one-to-one property. y log n b = log n M  Apply the power rule for logarithms. y = log n M log n b Isolate y. log b M = log n M log n b Substitute for y. log n ( b y ) = log n M Apply the one-to-one property. y log n b = log n M  Apply the power rule for logarithms. y = log n M log n b Isolate y. log b M = log n M log n b Substitute for y.

For example, to evaluate log 5 36 log 5 36 using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.

log 5 36 = log( 36 ) log( 5 ) Apply the change of base formula using base 10. 2.2266 Use a calculator to evaluate to 4 decimal places. log 5 36 = log( 36 ) log( 5 ) Apply the change of base formula using base 10. 2.2266 Use a calculator to evaluate to 4 decimal places.

The Change-of-Base Formula

The change-of-base formula can be used to evaluate a logarithm with any base.

For any positive real numbers M,b, M,b, and n, n, where n1  n1  and b1, b1,

log b M= log n M log n b . log b M= log n M log n b .

It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.

log b M= lnM lnb log b M= lnM lnb

and

log b M= logM logb log b M= logM logb

How To

Given a logarithm with the form log b M, log b M, use the change-of-base formula to rewrite it as a quotient of logs with any positive base n, n, where n1. n1.

  1. Determine the new base n, n, remembering that the common log, log( x ), log( x ), has base 10, and the natural log, ln( x ), ln( x ), has base e. e.
  2. Rewrite the log as a quotient using the change-of-base formula
    1. The numerator of the quotient will be a logarithm with base n n and argument M. M.
    2. The denominator of the quotient will be a logarithm with base n n and argument b. b.

Example 13

Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

Change log 5 3 log 5 3 to a quotient of natural logarithms.

Try It #13

Change log 0.5 8 log 0.5 8 to a quotient of natural logarithms.

Q&A

Can we change common logarithms to natural logarithms?

Yes. Remember that log9 log9 means log 10 9. log 10 9. So, log9= ln9 ln10 . log9= ln9 ln10 .

Example 14

Using the Change-of-Base Formula with a Calculator

Evaluate log 2 (10) log 2 (10) using the change-of-base formula with a calculator.

Try It #14

Evaluate log 5 (100) log 5 (100) using the change-of-base formula.

Media

Access these online resources for additional instruction and practice with laws of logarithms.

4.5 Section Exercises

Verbal

1.

How does the power rule for logarithms help when solving logarithms with the form log b ( x n )? log b ( x n )?

2.

What does the change-of-base formula do? Why is it useful when using a calculator?

Algebraic

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

3.

log b ( 7x2y ) log b ( 7x2y )

4.

ln( 3ab5c ) ln( 3ab5c )

5.

log b ( 13 17 ) log b ( 13 17 )

6.

log 4 ( x z w ) log 4 ( x z w )

7.

ln( 1 4 k ) ln( 1 4 k )

8.

log 2 ( y x ) log 2 ( y x )

For the following exercises, condense to a single logarithm if possible.

9.

ln( 7 )+ln( x )+ln( y ) ln( 7 )+ln( x )+ln( y )

10.

log 3 (2)+ log 3 (a)+ log 3 (11)+ log 3 (b) log 3 (2)+ log 3 (a)+ log 3 (11)+ log 3 (b)

11.

log b (28) log b (7) log b (28) log b (7)

12.

ln( a )ln( d )ln( c ) ln( a )ln( d )ln( c )

13.

log b ( 1 7 ) log b ( 1 7 )

14.

1 3 ln( 8 ) 1 3 ln( 8 )

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

15.

log( x 15 y 13 z 19 ) log( x 15 y 13 z 19 )

16.

ln( a −2 b −4 c 5 ) ln( a −2 b −4 c 5 )

17.

log( x 3 y 4 ) log( x 3 y 4 )

18.

ln( y y 1y ) ln( y y 1y )

19.

log( x 2 y 3 x 2 y 5 3 ) log( x 2 y 3 x 2 y 5 3 )

For the following exercises, condense each expression to a single logarithm using the properties of logarithms.

20.

log( 2 x 4 )+log( 3 x 5 ) log( 2 x 4 )+log( 3 x 5 )

21.

ln(6 x 9 )ln(3 x 2 ) ln(6 x 9 )ln(3 x 2 )

22.

2log(x)+3log(x+1) 2log(x)+3log(x+1)

23.

log(x) 1 2 log(y)+3log(z) log(x) 1 2 log(y)+3log(z)

24.

4 log 7 ( c )+ log 7 ( a ) 3 + log 7 ( b ) 3 4 log 7 ( c )+ log 7 ( a ) 3 + log 7 ( b ) 3

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.

25.

log 7 ( 15 ) log 7 ( 15 ) to base e e

26.

log 14 ( 55.875 ) log 14 ( 55.875 ) to base 10 10

For the following exercises, suppose log 5 ( 6 )=a log 5 ( 6 )=a and log 5 ( 11 )=b. log 5 ( 11 )=b. Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of a a and b. b. Show the steps for solving.

27.

log 11 ( 5 ) log 11 ( 5 )

28.

log 6 ( 55 ) log 6 ( 55 )

29.

log 11 ( 6 11 ) log 11 ( 6 11 )

Numeric

For the following exercises, use properties of logarithms to evaluate without using a calculator.

30.

log 3 ( 1 9 )3 log 3 ( 3 ) log 3 ( 1 9 )3 log 3 ( 3 )

31.

6 log 8 ( 2 )+ log 8 ( 64 ) 3 log 8 ( 4 ) 6 log 8 ( 2 )+ log 8 ( 64 ) 3 log 8 ( 4 )

32.

2 log 9 ( 3 )4 log 9 ( 3 )+ log 9 ( 1 729 ) 2 log 9 ( 3 )4 log 9 ( 3 )+ log 9 ( 1 729 )

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

33.

log 3 ( 22 ) log 3 ( 22 )

34.

log 8 ( 65 ) log 8 ( 65 )

35.

log 6 ( 5.38 ) log 6 ( 5.38 )

36.

log 4 ( 15 2 ) log 4 ( 15 2 )

37.

log 1 2 ( 4.7 ) log 1 2 ( 4.7 )

Extensions

38.

Use the product rule for logarithms to find all x x values such that log 12 ( 2x+6 )+ log 12 ( x+2 )=2. log 12 ( 2x+6 )+ log 12 ( x+2 )=2. Show the steps for solving.

39.

Use the quotient rule for logarithms to find all x x values such that log 6 ( x+2 ) log 6 ( x3 )=1. log 6 ( x+2 ) log 6 ( x3 )=1. Show the steps for solving.

40.

Can the power property of logarithms be derived from the power property of exponents using the equation b x =m? b x =m? If not, explain why. If so, show the derivation.

41.

Prove that log b ( n )= 1 log n ( b ) log b ( n )= 1 log n ( b ) for any positive integers b>1 b>1 and n>1. n>1.

42.

Does log 81 ( 2401 )= log 3 ( 7 )? log 81 ( 2401 )= log 3 ( 7 )? Verify the claim algebraically.

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