5.3 Use the Properties of Logarithms

Use the Properties of Logarithms

Now that we have learned about exponential and logarithmic functions, we can introduce some of the properties of logarithms. These will be very helpful as we continue to solve both exponential and logarithmic equations.

The first two properties derive from the definition of logarithms. Since [latex]{a}^{0}=1[/latex], we can convert this to logarithmic form and get [latex]{\text{log}}_{a}1=0[/latex]. Also, since [latex]{a}^{1}=a[/latex], we get [latex]{\text{log}}_{a}a=1[/latex].

In the next example we could evaluate the logarithm by converting to exponential form, as we have done previously, but recognizing and then applying the properties saves time.

The next two properties can also be verified by converting them from exponential form to logarithmic form, or the reverse.

The exponential equation [latex]{a}^{{\text{log}}_{a}x}=x[/latex] converts to the logarithmic equation [latex]{\text{log}}_{a}x={\text{log}}_{a}x[/latex], which is a true statement for positive values for [latex]x[/latex] only.

The logarithmic equation [latex]{\text{log}}_{a}{a}^{x}=x[/latex] converts to the exponential equation [latex]{a}^{x}={a}^{x}[/latex], which is also a true statement.

These two properties are called inverse properties because, when we have the same base, raising to a power “undoes” the log and taking the log “undoes” raising to a power. These two properties show the composition of functions. Both ended up with the identity function which shows again that the exponential and logarithmic functions are inverse functions.

In the next example, apply the inverse properties of logarithms.

There are three more properties of logarithms that will be useful in our work. We know exponential functions and logarithmic function are very interrelated. Our definition of logarithm shows us that a logarithm is the exponent of the equivalent exponential. The properties of exponents have related properties for exponents.

In the Product Property of Exponents, [latex]{a}^{m}·{a}^{n}={a}^{m+n}[/latex], we see that to multiply the same base, we add the exponents. The Product Property of Logarithms, [latex]{\text{log}}_{a}M·N={\text{log}}_{a}M+{\text{log}}_{a}N[/latex] tells us to take the log of a product, we add the log of the factors.

Product Property of Logarithms

We use this property to write the log of a product as a sum of the logs of each factor.

Similarly, in the Quotient Property of Exponents, [latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex], we see that to divide the same base, we subtract the exponents. The Quotient Property of Logarithms, [latex]{\text{log}}_{a}\frac{M}{N}={\text{log}}_{a}M-{\text{log}}_{a}N[/latex] tells us to take the log of a quotient, we subtract the log of the numerator and denominator.

Quotient Property of Logarithms

Note that [latex]{\text{log}}_{a}M-{\text{log}}_{a}N\ne {\text{log}}_{a}\left(M-N\right)[/latex].

We use this property to write the log of a quotient as a difference of the logs of each factor.

The third property of logarithms is related to the Power Property of Exponents, [latex]{\left({a}^{m}\right)}^{n}={a}^{m·n}[/latex], we see that to raise a power to a power, we multiply the exponents. The Power Property of Logarithms, [latex]{\text{log}}_{a}{M}^{p}=p\phantom{\rule{0.2em}{0ex}}{\text{log}}_{a}M[/latex] tells us to take the log of a number raised to a power, we multiply the power times the log of the number.

We use this property to write the log of a number raised to a power as the product of the power times the log of the number. We essentially take the exponent and throw it in front of the logarithm.

We summarize the Properties of Logarithms here for easy reference. While the natural logarithms are a special case of these properties, it is often helpful to also show the natural logarithm version of each property.

Now that we have the properties we can use them to “expand” a logarithmic expression. This means to write the logarithm as a sum or difference and without any powers.

We generally apply the Product and Quotient Properties before we apply the Power Property.

When we have a radical in the logarithmic expression, it is helpful to first write its radicand as a rational exponent.

The opposite of expanding a logarithm is to condense a sum or difference of logarithms that have the same base into a single logarithm. We again use the properties of logarithms to help us, but in reverse.

To condense logarithmic expressions with the same base into one logarithm, we start by using the Power Property to get the coefficients of the log terms to be one and then the Product and Quotient Properties as needed.

Use the Change-of-Base Formula

To evaluate a logarithm with any other base, we can use the Change-of-Base Formula. We will show how this is derived.

[latex]\begin{array}{cccccc}\text{Suppose we want to evaluate}\phantom{\rule{0.2em}{0ex}}{\text{log}}_{a}M.\hfill & & & \phantom{\rule{2em}{0ex}}{\text{log}}_{a}M\hfill & & \\ \text{Let}\phantom{\rule{0.2em}{0ex}}y={\text{log}}_{a}M.\hfill & & & \hfill \phantom{\rule{2em}{0ex}}y& =\hfill & {\text{log}}_{a}M\hfill \\ \text{Rewrite the expression in exponential form.}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}{a}^{y}& =\hfill & M\hfill \\ \text{Take the}\phantom{\rule{0.2em}{0ex}}{\text{log}}_{b}\phantom{\rule{0.2em}{0ex}}\text{of each side.}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}{\text{log}}_{b}{a}^{y}& =\hfill & {\text{log}}_{b}M\hfill \\ \text{Use the Power Property.}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}y{\text{log}}_{b}a& =\hfill & {\text{log}}_{b}M\hfill \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}y.\hfill & & & \hfill \phantom{\rule{2em}{0ex}}y& =\hfill & \frac{{\text{log}}_{b}M}{{\text{log}}_{b}a}\hfill \\ \text{Substitute}\phantom{\rule{0.2em}{0ex}}y={\text{log}}_{a}M.\hfill & & & \hfill \phantom{\rule{2em}{0ex}}{\text{log}}_{a}M& =\hfill & \frac{{\text{log}}_{b}M}{{\text{log}}_{b}a}\hfill \end{array}[/latex]

The Change-of-Base Formula introduces a new base [latex]b[/latex]. This can be any base [latex]b[/latex] we want where [latex]b \gt 0,b\ne 1[/latex]. Because our calculators have keys for logarithms base [latex]10[/latex] and base [latex]e[/latex], we will rewrite the Change-of-Base Formula with the new base as [latex]10[/latex] or [latex]e[/latex].

When we use a calculator to find the logarithm value, we usually round to three decimal places. This gives us an approximate value and so we use the approximately equal symbol [latex]\text{(≈)}[/latex].