Expand and Condense Logarithms

Learning Outcomes

• Combine the product, power, and quotient rules to expand logarithmic expressions
• Combine the product, power, and quotient rules to condense logarithmic expressions

Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.” Sometimes we apply more than one rule in order to simplify an expression. For example:

$\begin{array}{c}{\mathrm{log}}_{b}\left(\frac{6x}{y}\right)\hfill & ={\mathrm{log}}_{b}\left(6x\right)-{\mathrm{log}}_{b}y\hfill \\ \hfill & ={\mathrm{log}}_{b}6+{\mathrm{log}}_{b}x-{\mathrm{log}}_{b}y\hfill \end{array}$

We can also use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal (fraction) has a negative power:

$\begin{array}{c}{\mathrm{log}}_{b}\left(\frac{A}{C}\right)\hfill & ={\mathrm{log}}_{b}\left(A{C}^{-1}\right)\hfill \\ \hfill & ={\mathrm{log}}_{b}\left(A\right)+{\mathrm{log}}_{b}\left({C}^{-1}\right)\hfill \\ \hfill & ={\mathrm{log}}_{b}A+\left(-1\right){\mathrm{log}}_{b}C\hfill \\ \hfill & ={\mathrm{log}}_{b}A-{\mathrm{log}}_{b}C\hfill \end{array}$

We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.

Remember that we can only apply laws of logs to products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm. Consider the following example

$\begin{array}{c}\mathrm{log}\left(10+100\right)\overset{?}{=}\end{array}\mathrm{log}\left(10\right)+\mathrm{log}\left(100\right)\\\mathrm{log}\left(110\right)\overset{?}{=}1+2\\2.04\ne3$

Be careful to only apply the product rule when a logarithm has an argument that is a product or when you have a sum of logarithms.

In our first example, we will show that a logarithmic expression can be expanded by combining several of the rules of logarithms.

Example

Rewrite $\mathrm{ln}\left(\frac{{x}^{4}y}{7}\right)$ as a sum or difference of logs.

We can also use the rules for logarithms to simplify the logarithm of a radical expression.

Example

Expand $\mathrm{log}\left(\sqrt{x}\right)$.

Can we expand $\mathrm{ln}\left({x}^{2}+{y}^{2}\right)$? Use the space below to develop an argument one way or the other before you look at the solution.

Let us do one more example with an expression that contains several different mathematical operations.

Example

Expand ${\mathrm{log}}_{6}\left(\frac{64{x}^{3}\left(4x+1\right)}{\left(2x - 1\right)}\right)$.

In the following video, we show another example of expanding logarithms.

Condense Logarithms

We can use the rules of logarithms we just learned to condense sums and differences with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing.

Example

Write ${\mathrm{log}}_{3}\left(5\right)+{\mathrm{log}}_{3}\left(8\right)-{\mathrm{log}}_{3}\left(2\right)$ as a single logarithm.

In our next example, we show how to simplify a more complex logarithm by condensing it.

Example

Condense ${\mathrm{log}}_{2}\left({x}^{2}\right)+\frac{1}{2}{\mathrm{log}}_{2}\left(x - 1\right)-3{\mathrm{log}}_{2}\left({\left(x+3\right)}^{2}\right)$.

More examples of condensing logarithms are in the following video.

Summary

Given a sum or difference of logarithms with the same base, we can write an equivalent expression as a single logarithm.

1. Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power.
2. Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.
3. Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.