Logarithmic Functions

Relating Exponents to Logarithms

Logarithms and Exponents

Finding a logarithm means finding the exponent to which a base must be raised to equal a given value.
An exponent is a number that indicates how many times a base is multiplied by itself and shown by a raised number, such as:
43=4444^3=4\cdot 4\cdot 4
In the exponential equation y=bxy=b^x, bb is the base, xx is the exponent, and yy is the value of the power bxb^x. This equation can be read as "yy equals bb to the power of xx." An example of an exponential equation in the same form is:
8=2x8 = 2^x
It can be read as "8 equals 2 to the power of xx." Solving the equation involves finding the value of the exponent xx to which the base 2 must be raised to equal 8. The equation, and others like it, can be solved by finding a logarithm. A logarithm is the exponent xx to which a base bb is raised to produce a given number yy, written as:
x=logbyx=\log_b{y}
The given number yy is called the argument of the logarithm. The logarithmic equation can be read as "xx equals the logarithm base bb of yy." It is equivalent to the exponential equation, where b>0b>0 and b1b\neq 1:
y=bxy=b^{x}
Exponential Form Logarithmic Form
y=bxy=b^x
x=logbyx=\log_b{y}
8=2x8=2^x
x=log28x=\log_2{8}
8=238=2^3
3=log283=\log_2{8}

Some types of logarithms are used frequently. One is the common logarithm. A common logarithm is a logarithm with base 10, or the exponent to which 10 is raised to equal a given number; written as logx\log{x}, where xx is the given number. Notice that a common logarithm can be written without a subscript indicating the base. For example, log100\log{100} is the same as log10100\log_{10}{100}.

The irrational number ee is used frequently in logarithmic and exponential applications. Note that ee represents a constant and not a variable. The number ee is a nonterminating, nonrepeating decimal with a value of approximately 2.718281845.... A natural logarithm is a logarithm with base ee, or the exponent to which ee must be raised to equal a given number. A natural logarithm is written as lnx\ln{x}, where xx is the given number. So, ln2\ln{2} is the same as loge2\log_e{2}.

Step-By-Step Example
Evaluating Logarithms
Evaluate log1,000\log{1\text{,}000}.
Step 1

Rewrite log1,000\log{1\rm{,}000} in exponential form.

Since there is no base indicated, the expression is a common logarithm, meaning that the base is 10:
log1,000=log101,000\log{1\text{,}000}=\log_{10}{1\text{,}000}
Let xx represent the exponent, and convert the equation from the form logby=x\log_b{y}=x to bx=yb^x=y.
log101,000=x10x=1,000\begin{aligned}\log_{10}{1\text{,}000}&=x\\10^x&=1\text{,}000\end{aligned}
Step 2
Calculate the value of xx.
10x=1,000103=1,000\begin{aligned}10^x&=1\text{,}000\\10^3&=1\text{,}000\end{aligned}
The value of xx is 3.
Solution
Since the value of xx is 3, log1,000\log{1\text{,}000} is equal to 3:
log1,000=3\log{1\text{,}000}=3

Technology is often required to evaluate logarithms. Since the base of a logarithm is restricted to positive real numbers other than 1, the number of possible logarithm bases is infinite. Common calculators come with two logarithmic functions: log for the common logarithm and ln for the natural logarithm. To evaluate log900\log{900}, enter log(900), which gives an approximate result of 2.95. To evaluate ln900\ln{900}, enter ln(900), which gives an approximate answer of 6.80. Logarithms in other bases must be converted to either base 10 or base ee to be evaluated with a calculator.

Inverses of Exponential Functions

A logarithmic function is the inverse of an exponential function.

An input is a value in the domain of a function. An output is a value in the range of a function. For an exponential function of the form f(x)=bxf(x)=b^x, the inputs of the function are exponents and the outputs are powers of the base bb.

An inverse function is the result of switching the inputs and outputs of a given function when the result is also a function. The composition of a function and its inverse is the identity function. If the inputs and outputs of the exponential f(x)=bxf(x)=b^x are switched to produce an inverse function, the inputs of the inverse function will be powers of the base bb, and the outputs will be exponents. In other words, the rule for the inverse function will be a logarithm.

A logarithmic function, where b>0b>0 and b1b\neq 1, has the form:
f(x)=logbxf(x)=\log_b{x}
A logarithmic function f(x)=logbxf(x)=\log_b{x} is the inverse of the exponential function f(x)=bxf(x)=b^x.
The inverse function of an exponential function is its logarithmic function. For example, when 3 is the input for the exponential function f(x)=2xf(x)=2^x, the output is 8. For the inverse logarithmic function f(x)=log2xf(x)=\log_2{x}, the relationship is reversed. When 8 is the input, the output is 3.
Step-By-Step Example
Using the Inverse Function to Graph a Logarithmic Function
Graph the given logarithmic function and its inverse, and compare the graphs:
f(x)=log2xf(x)=\log_2{x}
Step 1

Write the equation of the inverse exponential function.

The inverse of the given logarithmic function is the exponential function:

f(x)=2xf(x)=2^x
Step 2

Complete a table of values for the inverse exponential function.

xx
f(x)=2xf(x)=2^x
2-2
0.250.25
1-1
0.50.5
00
11
11
22
22
44
Step 3

Complete a table of values for the logarithmic function by switching the input and output values for the inverse exponential function.

xx
f(x)=log2xf(x)=\log_2{x}
0.250.25
2-2
0.50.5
1-1
11
00
22
11
44
22
Solution

Plot ordered pairs from the table for the inverse exponential function:

f(x)=2xf(x)=2^x

Then, connect them with a smooth curve.

Plot ordered pairs from the table for the logarithmic function:

f(x)=log2xf(x)=\log_2{x}

Then, connect them with a smooth curve.

The curves are reflections of each other across the diagonal line f(x)=xf(x)=x.