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 \(\PageIndex{1}\), to better estimate the solution.
Figure \(\PageIndex{2}\)
Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in Figure \(\PageIndex{2}\) 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.
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 \({\log}_232=5\). 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:
\[\begin{align} \log_b(x)=y\Leftrightarrow b^y=x, b> 0, b\neq 1 \end{align}\]
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 \(\log_bx\). 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 \(y={\log}^b(x)\) and \(y=b^x\) are inverse functions.
A logarithm base \(b\) of a positive number \(x\) satisfies the following definition.
For \(x>0\), \(b>0\), \(b≠1\),
\[\begin{align} y={\log}_b(x)\text{ is equivalent to } b^y=x \end{align}\]
where,
- we read \({\log}_b(x)\) as, “the logarithm with base \(b\) of \(x\)” or the “log base \(b\) of \(x\)."
- the logarithm \(y\) is the exponent to which \(b\) must be raised to get \(x\).
Also, since the logarithmic and exponential functions switch the \(x\) and \(y\) values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,
- the domain of the logarithm function with base \(b\) is \((0,\infty)\).
- the range of the logarithm function with base \(b\) is \((−\infty,\infty)\).
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 . 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.
A common logarithm is a logarithm with base \(10\). We write \({\log}_{10}(x)\) simply as \(\log(x)\). The common logarithm of a positive number \(x\) satisfies the following definition.
For \(x>0\),
\[\begin{align} y={\log}(x)\text{ is equivalent to } {10}^y=x \end{align}\]
We read \(\log(x)\) as, “the logarithm with base \(10\) of \(x\) ” or “log base \(10\) of \(x\).”
The logarithm \(y\) is the exponent to which \(10\) must be raised to get \(x\).
- Rewrite the argument \(x\) as a power of \(10\): \({10}^y=x\).
- Use previous knowledge of powers of \(10\) to identify \(y\) by asking, “To what exponent must \(10\) be raised in order to get \(x\)?”
Evaluate \(y=\log(1000)\) without using a calculator.
Solution
First we rewrite the logarithm in exponential form: \({10}^y=1000\). Next, we ask, “To what exponent must \(10\) be raised in order to get \(1000\)?” We know
\({10}^3=1000\)
Therefore, \(\log(1000)=3\).
Evaluate \(y=\log(1,000,000)\).
- Answer
-
\(\log(1,000,000)=6\)
- Press [LOG] .
- Enter the value given for \(x\),followed by [ ) ] .
- Press [ENTER] .
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 base \(e\) logarithm, \({\log}_e(x)\), has its own notation,\(\ln(x)\). 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, \(\ln1=0\). 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.
A natural logarithm is a logarithm with base \(e\). We write \({\log}_e(x)\) simply as \(\ln(x)\). The natural logarithm of a positive number \(x\) satisfies the following definition.
For \(x>0\),
\(y=\ln(x)\) is equivalent to \(e^y=x\)
We read \(\ln(x)\) as, “the logarithm with base \(e\) of \(x\)” or “the natural logarithm of \(x\).”
The logarithm \(y\) is the exponent to which \(e\) must be raised to get \(x\).
Since the functions \(y=e^x\) and \(y=\ln(x)\) are inverse functions, \(\ln(e^x)=x\) for all \(x\) and \(e^{\ln (x)}=x\) for \(x>0\).