4.3 Logarithmic Functions

Converting from Logarithmic to Exponential Form

In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is ${10}^x=500,$ where $x$ represents the difference in magnitudes on the Richter Scale. How would we solve for $x?$

We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve ${10}^x=500.$ We know that ${10}^2=100$ and ${10}^3=1000,$ so it is clear that $x$ must be some value between 2 and 3, since $y={10}^x$ is increasing. We can examine a graph, as in Figure 2, to better estimate the solution.

Figure 2

Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in Figure 2 passes the horizontal line test. The exponential function $y=b^x$ is one-to-one, so its inverse, $x=b^y$ is also a function. As is the case with all inverse functions, we simply interchange $x$ and $y$ and solve for $y$ to find the inverse function. To represent $y$ as a function of $x,$ we use a logarithmic function of the form $y={\log }_b\left(x\right).$ The base $b$ logarithm of a number is the exponent by which we must raise $b$ to get that number.

We read a logarithmic expression as, “The logarithm with base $b$ of $x$ is equal to $y,$ ” or, simplified, “log base $b$ of $x$ is $y.$ ” We can also say, “ $b$ raised to the power of $y$ is $x,$ ” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since $2^5=32,$ we can write ${\log }_{2}32=5.$ We read this as “log base 2 of 32 is 5.”

We can express the relationship between logarithmic form and its corresponding exponential form as follows:

Note that the base $b$ is always positive.

Because logarithm is a function, it is most correctly written as ${\log }_b(x),$ using parentheses to denote function evaluation, just as we would with $f(x).$ However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as ${\log }_{b}x.$ Note that many calculators require parentheses around the $x.$

We can illustrate the notation of logarithms as follows:

Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means $y={\log }_b\left(x\right)$ and $y=b^x$ are inverse functions.

Converting from Exponential to Logarithmic Form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base $b,$ exponent $x,$ and output $y.$ Then we write $x={\log }_b\left(y\right).$

Evaluating Logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider ${\log }_{2}8.$ We ask, “To what exponent must $2$ be raised in order to get 8?” Because we already know $2^3=8,$ it follows that ${\log }_{2}8=3.$

Now consider solving ${\log }_{7}49$ and ${\log }_{3}27$ mentally.

• We ask, “To what exponent must 7 be raised in order to get 49?” We know $7^2=49.$ Therefore, ${\log }_{7}49=2$
• We ask, “To what exponent must 3 be raised in order to get 27?” We know $3^3=27.$ Therefore, ${\log }_{3}27=3$

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate ${\log }_{\frac{2}{3}}\frac{4}{9}$ mentally.

• We ask, “To what exponent must $\frac{2}{3}$ be raised in order to get $\frac{4}{9}?$ ” We know $2^2=4$ and $3^2=9,$ so ${\left(\frac{2}{3}\right)}^2=\frac{4}{9}.$ Therefore, ${\log }_{\frac{2}{3}}\left(\frac{4}{9}\right)=2.$

Using Common Logarithms

Sometimes you may see a logarithm written without a base. When you see one written this way, you need to look at the expression before evaluating it. It may be that the base you use doesn't matter. If you find it in computer science, it often means ${\log }_2(x)$. However, in mathematics it almost always means the common logarithm of 10. In other words, the expression $\log (x)$ often means ${\log }_{10}(x).$

Currently, we use ${\log }_b\left(x\right),\lg \left(x\right)$ as the common logarithm, $\mathrm{lb}\left(x\right)$ as the binary logarithm, and $\ln \left(x\right)$ as the natural logarithm. Writing $\lg \left(x\right)$ without specifying a base is now considered bad form, despite being frequently found in older materials.

Using Natural Logarithms

The most frequently used base for logarithms is $e,$ the value of which is approximately $2.71828$. Base $e$ logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base $e$ logarithm, ${\log }_e\left(x\right),$ has its own notation, $\ln (x).$

Most values of $\ln \left(x\right)$ can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, $\ln 1=0.$ For other natural logarithms, we can use the $\ln$ key that can be found on most scientific calculators. We can also find the natural logarithm of any power of $e$ using the inverse property of logarithms.