Lane ORCCA (2020–2021): Open Resources for Community College Algebra

Subsection 14.2.1 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 \(10^x=500\text{,}\) where \(x\) represents the difference in magnitudes on the Richter Scale. How would we solve for\(x\text{?}\)

We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve \(10^x=500\text{.}\) We know that \(10^2=100\) and \(10^3=1000\text{,}\) so it is clear that \(x\) must be some value between 2 and 3, since, as we learned in Section 14.1, \(y=10^x\) is an increasing function.

To find an algebraic solution, we must introduce a new function. To represent \(y\) as a function of \(x\text{,}\) we use a logarithmic function of the form \(y=\mathrm{log}_{b}\left(x\right)\text{.}\) 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 \(y\text{.}\)” 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 \(2^5=32\text{,}\) we can write \(\mathrm{log}_{2}32=5\text{.}\) We read this as “log base \(2\) of \(32\) is \(5\text{.}\)”

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

Note that the base \(b\) is always positive.

Because a logarithm is a function, it is most correctly written as \(\mathrm{log}_{b}\left(x\right)\text{,}\) using parentheses to denote function evaluation, just as we would with \(f\left(x\right)\text{.}\) 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 \(\mathrm{log}_{b}x\text{.}\) Note that many calculators require parentheses around the \(x\text{.}\)

We can illustrate the notation of logarithms as follows:

The following diagram may help you identify the base and exponent when converting between exponential and logarithmic forms of an equation:

1. Examine the equation \(y=\mathrm{log}_{b}\left(x\right)\) and identify \(b\text{,}\) \(y\text{,}\) and \(x\text{.}\)

2. Rewrite \(\mathrm{log}_{b}\left(x\right)=y\) as \(b^y=x\text{.}\)

Subsection 14.2.2 Converting from Exponential to Logarithmic Form

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

Subsection 14.2.3 Evaluating Logarithms

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

Now consider solving \(\mathrm{log}_{7}49\) and \(\mathrm{log}_{3}27\) mentally.

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

• We ask, “To what exponent must 3 be raised in order to get 27?” We know \(3^3=27\text{.}\) Therefore, \(\mathrm{log}_{3}27=3\text{.}\)

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

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

Subsection 14.2.4 Using Common Logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression \(\mathrm{log}\left(x\right)\) means \(\mathrm{log}_{10}\left(x\right)\text{.}\) We call a base-10 logarithm a common logarithm. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

Subsection 14.2.5 Using Natural Logarithms

The most frequently used base for logarithms is \(e\text{.}\) Base \(e\) logarithms are important in calculus and many scientific applications; they are called natural logarithms. The base \(e\) logarithm, \(\mathrm{log}_{e}\left(x\right)\text{,}\) has its own shorthand notation, \(\mathrm{ln}\left(x\right)\text{.}\)

Most values of \(\mathrm{ln}\left(x\right)\) can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 for any base, \(\mathrm{ln}1=0\text{.}\) For other natural logarithms, we can use the \(\mathrm{ln}\) key that can be found on most scientific calculators. We can also find the natural logarithm of any power of \(e\) using what we've learned so far about the relationship between logarithm and exponential equations.