13.3: 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 \(\PageIndex{2}\), 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 in Figure \(\PageIndex{2}\) 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 \(y = \log_b(x)\). The base \(b\) logarithm of a number is the exponent by which we must raise \(b\) to get that number.

It is critical to recognize that 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:\[\log_b(x) = y \iff b^y = x, b > 0, b \neq 1.\nonumber\]The symbol, \(\iff\), means "if and only if." Since \(b\) is the base of an exponential function, \(b\) is always positive.

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 \(y = \log_b(x)\) and \(y = b^x\) are inverse functions. This leads us to quickly develop some fundamental theory about logarithmic functions.

In fact, it is due to the inverse nature of the logarithmic and exponential functions that \(x > 0\), \(b > 0\), and \(b \neq 1\) are restrictions for the logarithmic function. Since \(y = \log_b(x)\) is the same as \(b^y = x\), and we know the base of an exponential function must be positive, \(b > 0\). Moreover, when defining the exponential function, we took great care to state that the base could not be \(1\). Thus, \(b \neq 1\). Finally, exponential functions only return positive values. Hence, \(x > 0\).

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 \(x = \log_b(y)\).

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 simplifying \(\log_7(49)\) 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 \(\log_{2/3} \left( \frac{4}{9} \right)\) 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, \(\log_{2/3} \left( \frac{4}{9} \right) = 2\).

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 \(\log(x)\) means \(\log_{10}(x)\). We call a base-10 logarithm a common logarithm.

The common logarithm otherwise follows the standard definition of a logarithmic function: For \(x > 0\),\[y = \log(x) \text{ is equivalent to } 10^y = x.\nonumber\]Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars, decibel levels of sounds, and the pH of acids and bases also use common logarithms.

We can perform a quick "logic check" on our answer to Example \(\PageIndex{6}\). Note that \(10^2 = 100\) and that \(10^3 = 1000\). Since \(321\) is between \(100\) and \(1000\), we know that \(\log(321)\) must be between \(\log(100) = 2\) and \(\log(1000) = 3\).

Using Natural Logarithms

The most frequently used base for logarithms is \(e\). Base \(e\) logarithms are important in Calculus and some scientific applications; they are called natural logarithms.

The natural logarithm otherwise follows the standard definition of a logarithmic function: For \(x > 0\),\[y = \ln(x) \text{ is equivalent to } e^y = x.\nonumber\]Most values of \(\ln(x)\) 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 \(\mathrm{ln}\) 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.