Logarithmic Functions
Using our understanding of exponential functions, we can discuss their inverses, which are the logarithmic functions.
Inverse Functions
For any one-to-one function [latex]f\left(x\right)=y[/latex], a function [latex]{f}^{-1}\left(x\right)[/latex] is an inverse function of [latex]f[/latex] if [latex]{f}^{-1}\left(y\right)=x[/latex].
The notation [latex]{f}^{-1}[/latex] is read “[latex]f[/latex] inverse.” Like any other function, we can use any variable name as the input for [latex]{f}^{-1}[/latex], so we will often write [latex]{f}^{-1}\left(x\right)[/latex], which we read as [latex]"f[/latex] inverse of [latex]x[/latex]“.
Logarithmic functions come in handy when we need to consider any phenomenon that varies over a wide range of values, such as pH in chemistry or decibels in sound levels.
The exponential function [latex]f(x)=b^x[/latex] is one-to-one, with domain [latex](−\infty ,\infty)[/latex] and range [latex](0,\infty )[/latex]. Therefore, it has an inverse function, called the logarithmic function with base [latex]b[/latex].
For any [latex]b>0, \, b \ne 1[/latex], the logarithmic function with base [latex]b[/latex], denoted [latex]\log_b[/latex], has domain [latex](0,\infty )[/latex] and range [latex](−\infty ,\infty )[/latex], and satisfies
[latex]\log_b(x)=y[/latex] if and only if [latex]b^y=x[/latex].
logarithmic functions
A logarithmic function is the inverse of an exponential function and is written as [latex]log_{b}(x)[/latex]. For a given base [latex]b[/latex], it tells us the power to which [latex]b[/latex] must be raised to get [latex]x[/latex].
The most commonly used logarithmic function is the function [latex]\log_e (x)[/latex]. Since this function uses natural [latex]e[/latex] as its base, it is called the natural logarithm. Here we use the notation [latex]\ln(x)[/latex] or [latex]\ln x[/latex] to mean [latex]\log_e (x)[/latex].
Euler’s number, denoted as [latex]e[/latex], is a fundamental mathematical constant approximately equal to [latex]2.71828[/latex]. It is the base of the natural logarithm and the natural exponential function, known for its unique properties in calculus, especially in relation to growth processes and compound interest calculations.
Before solving some equations involving exponential and logarithmic functions, let’s review the basic properties of logarithms.
Properties of Logarithms
If [latex]a,b,c>0, \, b\ne 1[/latex], and [latex]r[/latex] is any real number, then
[latex]\begin{array}{cccc}1.\phantom{\rule{2em}{0ex}}\log_b (ac)=\log_b (a)+\log_b (c)\hfill & & & \text{(Product property)}\hfill \\ 2.\phantom{\rule{2em}{0ex}}\log_b(\frac{a}{c})=\log_b (a) -\log_b (c)\hfill & & & \text{(Quotient property)}\hfill \\ 3.\phantom{\rule{2em}{0ex}}\log_b (a^r)=r \log_b (a)\hfill & & & \text{(Power property)}\hfill \end{array}[/latex]
Solve each of the following equations for [latex]x[/latex].
- [latex]\ln \left(\frac{1}{x}\right)=4[/latex]
- [latex]\log_{10} \sqrt{x}+ \log_{10} x=2[/latex]
- [latex]\ln(2x)-3 \ln(x^2)=0[/latex]
Solve each of the following equations for [latex]x[/latex].
- [latex]5^x=2[/latex]
- [latex]e^x+6e^{−x}=5[/latex]
In calculations involving logarithms, you might have noticed that calculators typically provide only the common logarithm ([latex]\log_{10}[/latex]) and natural logarithm (base [latex]e[/latex]). However, exponential functions and logarithm functions can be expressed in terms of any desired base [latex]b[/latex].
To work with different bases, we can use the change-of-base formula to convert to a base your calculator can handle.
change-of-base formulas
The change-of-base formula allows you to evaluate logarithms with any base using only the common or natural logarithm functions typically available on a calculator.
For any real number [latex]x[/latex] and bases [latex]a>0, \, b>0[/latex], and [latex]a\ne 1, \, b\ne 1[/latex], the exponential expression [latex]a^x[/latex]can be rewritten using a logarithm with base [latex]b[/latex] as [latex]a^x=b^{x \log_b a}[/latex].
The change of base formula is:
If [latex]b=e[/latex], this exponential expression reduces to [latex]a^x=e^{x \log_e a}=e^{x \ln a}[/latex]. The change of base formula reduces to [latex]\log_a x=\frac{\ln x}{\ln a}[/latex].
Proof
For the first change-of-base formula, we begin by making use of the power property of logarithmic functions. We know that for any base [latex]b>0, \, b\ne 1, \, \log_b (a^x)=x \log_b a[/latex]. Therefore,
In addition, we know that [latex]b^x[/latex] and [latex]\log_b (x)[/latex] are inverse functions. Therefore,
Combining these last two equalities, we conclude that [latex]a^x=b^{x \log_b a}[/latex].
To prove the second property, we show that
Let [latex]u=\log_b a, \, v=\log_a x[/latex], and [latex]w=\log_b x[/latex]. We will show that [latex]u·v=w[/latex]. By the definition of logarithmic functions, we know that [latex]b^u=a, \, a^v=x[/latex], and [latex]b^w=x[/latex]. From the previous equations, we see that
Therefore, [latex]b^{uv}=b^w[/latex]. Since exponential functions are one-to-one, we can conclude that [latex]u·v=w[/latex].
[latex]_\blacksquare[/latex]
How to: Use the Change-of-Base Formulas
- Take the logarithm you need to evaluate, [latex](\log_b a)[/latex]
- Using the change-of-base formula, rewrite it as:
[latex]\log_a x=\frac{\log_b x}{\log_b a}[/latex]
- Use your calculator to find the common log (base [latex]10[/latex]) of [latex]a[/latex],[latex]\log{a}[/latex], and the common log of [latex]x[/latex], [latex]\log{x}[/latex].
- Divide these two values to compute [latex]\log_a x[/latex]
Note: You can also use the natural logarithm ([latex]ln[/latex]) in place of the common logarithm ([latex]log[/latex]) if preferred.
Use a calculator to evaluate [latex]\log_3 7[/latex] using the change-of-base formula.