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 [latex]500[/latex] 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 [latex]{10}^{x}=500[/latex], where [latex]x[/latex] represents the difference in magnitudes on the Richter Scale. How would we solve for [latex]x[/latex]?

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

Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph above passes the horizontal line test. The exponential function [latex]y={b}^{x}[/latex] is one-to-one, so its inverse, [latex]x={b}^{y}[/latex] 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 [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]. 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 [latex]2[/latex] logarithm of [latex]32[/latex] is [latex]5[/latex], because [latex]5[/latex] is the exponent we must apply to [latex]2[/latex] to get [latex]32[/latex]. Since [latex]{2}^{5}=32[/latex], we can write [latex]{\mathrm{log}}_{2}32=5[/latex]. We read this as “log base [latex]2[/latex] of [latex]32[/latex] is [latex]5[/latex].”

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

Note that the base b is always positive.

Because logarithms are functions, they are most correctly written as [latex]{\mathrm{log}}_{b}\left(x\right)[/latex], using parentheses to denote function evaluation, just as we would with [latex]f\left(x\right)[/latex]. 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 [latex]{\mathrm{log}}_{b}x[/latex]. Note that many calculators require parentheses around the x.

The parentheses in a logarithmic expression may contain a single variable like x, or it may contain another expression like 2x-3.  We refer to the contents inside of the parentheses as the argument.  For example, in the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex], we say that x is the argument.  In the function [latex]y={\mathrm{log}}_{b}\left(2x-3\right)[/latex], the argument is [latex]2x-3[/latex].

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 [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] and [latex]y={b}^{x}[/latex] are inverse functions.

In our first example, we will convert logarithmic equations into exponential equations.

In the following video, we present more examples of rewriting logarithmic equations as exponential equations.

 Convert from Exponential to Logarithmic Form

To convert from exponential form to logarithmic form, we follow the same steps in reverse. We identify the base b, exponent x, and output y. Then we write [latex]x={\mathrm{log}}_{b}\left(y\right)[/latex].

In our last video, we show more examples of writing logarithmic equations as exponential equations.

Summary

The base b logarithm of a number is the exponent by which we must raise b to get that number. Logarithmic functions are the inverse of exponential functions, and it is often easier to understand them through this lens. We can never take the logarithm of a negative number, therefore [latex]{\mathrm{log}}_{b}\left(x\right)=y[/latex] is defined for [latex]b>0[/latex]