Jay Abramson; Sharon North

Key Concepts

6.1 Exponential Functions

An exponential function is defined as a function with a positive constant other than $1$ raised to a variable exponent. See Example 1.
A function is evaluated by solving at a specific value. See Example 2 and Example 3.
An exponential model can be found when the growth rate and initial value are known. See Example 4.
An exponential model can be found when the two data points from the model are known. See Example 5.
An exponential model can be found using two data points from the graph of the model. See Example 6.
An exponential model can be found using two data points from the graph and a calculator. See Example 7.
The value of an account at any time $t$ can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known. See Example 8.
The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known. See Example 9.
The number $e$ is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is $e \approx 2.718282.$
Scientific and graphing calculators have the key $[e^{x}]$ or $[\exp (x)]$ for calculating powers of $e .$ See Example 10.
Continuous growth or decay models are exponential models that use $e$ as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known. See Example 11 and Example 12.

6.2 Graphs of Exponential Functions

The graph of the function $f (x) = b^{x}$ has a y-intercept at $(0, 1),$ domain $(- \infty, \infty),$ range $(0, \infty),$ and horizontal asymptote $y = 0.$ See Example 1.
If $b > 1,$ the function is increasing. The left tail of the graph will approach the asymptote $y = 0,$ and the right tail will increase without bound.
If $0 < b < 1,$ the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote $y = 0.$
The equation $f (x) = b^{x} + d$ represents a vertical shift of the parent function $f (x) = b^{x} .$
The equation $f (x) = b^{x + c}$ represents a horizontal shift of the parent function $f (x) = b^{x} .$ See Example 2.
Approximate solutions of the equation $f (x) = b^{x + c} + d$ can be found using a graphing calculator. See Example 3.
The equation $f (x) = a b^{x},$ where $a > 0,$ represents a vertical stretch if $| a | > 1$ or compression if $0 < | a | < 1$ of the parent function $f (x) = b^{x} .$ See Example 4.
When the parent function $f (x) = b^{x}$ is multiplied by $- 1,$ the result, $f (x) = - b^{x},$ is a reflection about the x-axis. When the input is multiplied by $- 1,$ the result, $f (x) = b^{- x},$ is a reflection about the y-axis. See Example 5.
All translations of the exponential function can be summarized by the general equation $f (x) = a b^{x + c} + d .$ See Table 3.
Using the general equation $f (x) = a b^{x + c} + d,$ we can write the equation of a function given its description. See Example 6.

6.3 Logarithmic Functions

The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See Example 1.
Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See Example 2.
Logarithmic functions with base $b$ can be evaluated mentally using previous knowledge of powers of $b .$ See Example 3 and Example 4.
Common logarithms can be evaluated mentally using previous knowledge of powers of $10.$ See Example 5.
When common logarithms cannot be evaluated mentally, a calculator can be used. See Example 6.
Real-world exponential problems with base $10$ can be rewritten as a common logarithm and then evaluated using a calculator. See Example 7.
Natural logarithms can be evaluated using a calculator Example 8.

6.4 Graphs of Logarithmic Functions

To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for $x .$ See Example 1 and Example 2
The graph of the parent function f(x)= log b ( x ) f(x)= log b ( x ) has an x-intercept at ( 1,0 ), ( 1,0 ), domain ( 0,∞ ), ( 0,∞ ), range ( −∞,∞ ), ( −∞,∞ ), vertical asymptote x=0, x=0, and
- if $b > 1,$ the function is increasing.
- if $0 < b < 1,$ the function is decreasing.
See Example 3.
The equation f(x)= log b ( x+c ) f(x)= log b ( x+c ) shifts the parent function y= log b ( x ) y= log b ( x ) horizontally
- left $c$ units if $c > 0.$
- right $c$ units if $c < 0.$
See Example 4.
The equation f(x)= log b ( x )+d f(x)= log b ( x )+d shifts the parent function y= log b ( x ) y= log b ( x ) vertically
- up $d$ units if $d > 0.$
- down $d$ units if $d < 0.$
See Example 5.
For any constant a>0, a>0, the equation f(x)=a log b ( x ) f(x)=a log b ( x )
- stretches the parent function $y = \log_{b} (x)$ vertically by a factor of $a$ if $| a | > 1.$
- compresses the parent function $y = \log_{b} (x)$ vertically by a factor of $a$ if $| a | < 1.$
See Example 6 and Example 7.
When the parent function y= log b ( x ) y= log b ( x ) is multiplied by −1, −1, the result is a reflection about the x-axis. When the input is multiplied by −1, −1, the result is a reflection about the y-axis.
- The equation $f (x) = - \log_{b} (x)$ represents a reflection of the parent function about the x-axis.
- The equation $f (x) = \log_{b} (- x)$ represents a reflection of the parent function about the y-axis.
See Example 8.
- A graphing calculator may be used to approximate solutions to some logarithmic equations See Example 9.
All translations of the logarithmic function can be summarized by the general equation $f (x) = a \log_{b} (x + c) + d .$ See Table 4.
Given an equation with the general form $f (x) = a \log_{b} (x + c) + d,$ we can identify the vertical asymptote $x = - c$ for the transformation. See Example 10.
Using the general equation $f (x) = a \log_{b} (x + c) + d,$ we can write the equation of a logarithmic function given its graph. See Example 11.

6.5 Logarithmic Properties

We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See Example 1.
We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See Example 2.
We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. See Example 3, Example 4, and Example 5.
We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. See Example 6, Example 7, and Example 8.
The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. See Example 9, Example 10, Example 11, and Example 12.
We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. See Example 13.
The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and $e$ as the quotient of natural or common logs. That way a calculator can be used to evaluate. See Example 14.

6.6 Exponential and Logarithmic Equations

We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See Example 1.
When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See Example 2, Example 3, and Example 4.
When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See Example 5.
We can solve exponential equations with base $e,$ by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See Example 6 and Example 7.
After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See Example 8.
When given an equation of the form $\log_{b} (S) = c,$ where $S$ is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation $b^{c} = S,$ and solve for the unknown. See Example 9 and Example 10.
We can also use graphing to solve equations with the form $\log_{b} (S) = c .$ We graph both equations $y = \log_{b} (S)$ and $y = c$ on the same coordinate plane and identify the solution as the x-value of the intersecting point. See Example 11.
When given an equation of the form $\log_{b} S = \log_{b} T,$ where $S$ and $T$ are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation $S = T$ for the unknown. See Example 12.
Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See Example 13.

6.7 Exponential and Logarithmic Models

The basic exponential function is $f (x) = a b^{x} .$ If $b > 1,$ we have exponential growth; if $0 < b < 1,$ we have exponential decay.
We can also write this formula in terms of continuous growth as $A = A_{0} e^{k x},$ where $A_{0}$ is the starting value. If $A_{0}$ is positive, then we have exponential growth when $k > 0$ and exponential decay when $k < 0.$ See Example 1.
In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay. See Example 2.
We can find the age, $t,$ of an organic artifact by measuring the amount, $k,$ of carbon-14 remaining in the artifact and using the formula $t = \frac{\ln (k)}{- 0.000121}$ to solve for $t .$ See Example 3.
Given a substance’s doubling time or half-time, we can find a function that represents its exponential growth or decay. See Example 4.
We can use Newton’s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time. See Example 5.
We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors. See Example 6.
We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data. See Example 7.
Any exponential function with the form $y = a b^{x}$ can be rewritten as an equivalent exponential function with the form $y = A_{0} e^{k x}$ where $k = \ln b .$ See Example 8.

6.8 Fitting Exponential Models to Data

Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.
We use the command “ExpReg” on a graphing utility to fit function of the form $y = a b^{x}$ to a set of data points. See Example 1.
Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time.
We use the command “LnReg” on a graphing utility to fit a function of the form $y = a + b \ln (x)$ to a set of data points. See Example 2.
Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit.
We use the command “Logistic” on a graphing utility to fit a function of the form $y = \frac{c}{1 + a e^{- b x}}$ to a set of data points. See Example 3.