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Section 14.2 Expanding Logarithms

In this section, we will do the same thing we did in Section 14.1, but backwards: we will start with a bunch of stuff in one log, and we want to separate it into multiple logarithms.

Let's copy the table we had from Section 14.1, but since we are working the other way around, let's write the equations the other way around as well

Table 14.4. Log and Exponent Properties
Exponent Rule Logarithm Rule
\(b^{x+y}=b^xb^y\) \(\log_b(xy)=\log_b(x)+\log_b(y)\)
Add inside, multiply outside Multiply inside, add outside
\(b^{x-y}=\dfrac{b^x}{b^y}\) \(\log_b\left(\dfrac{x}{y}\right)=\log_b(x)-\log_b(y)\)
Subtract inside, divide outside Divide inside, subtract outside
\(b^{xy}=\left(b^x\right)^y\) \(\log_b(x^y)=y\log_b(x)\)
Multiply inside, exponent outside Exponent inside, multiply outside

Example 14.5.

Suppose we want to expand the following log as much as possible:

\begin{equation*} \log_7\left(\dfrac{x^5\sqrt{y}}{z^4}\right) \end{equation*}

It may be tempting to bring some of the exponents out, but we can only do that when the exponent is on the entire inside. So, for example, we cannot yet bring down the 5 because right now it's only the exponent on part of the inside (just the \(x\)). But, if we separate this into multiple logs, then we will have each exponent on its own piece. So, let's start by using the first two rules to separate the logs.

\begin{equation*} \log_7\left(x^5\right)+\log_7\left(\sqrt{y}\right)-\log_7\left(z^4\right) \end{equation*}

Now, we will want to bring our exponents outside. However, the middle term has a square root instead of an exponent. So, we need to convert \(\sqrt{y}\) into \(y\) to a power. Remember that the square root is the same thing as the \(1/2\) power

\begin{equation*} \log_7\left(x^5\right)+\log_7\left(y^{1/2}\right)-\log_7\left(z^4\right) \end{equation*}

Now we are finally ready to get our answer by bringing down the exponents.

\begin{equation*} 5\log_7\left(x\right)+\frac{1}{2}\log_7\left(y\right)-4\log_7\left(z\right) \end{equation*}

Checkpoint 14.6.

Separate the following as much as possible

\begin{equation*} \ln\left(\dfrac{x^{13}}{z^7\sqrt[6]{x+1}}\right) \end{equation*}
Answer.
\begin{equation*} 13\ln\left(x\right)-7\ln\left(z\right)-\frac{1}{6}\ln\left(x+1\right) \end{equation*}
Solution.

First, we need to use the first two rules to separate this into multiple logs.

\begin{equation*} \ln\left(x^{13}\right)-\ln\left(z^7\right)-\ln\left(\sqrt[6]{x+1}\right) \end{equation*}

We will want to bring down our exponents, but first we have to rewrite the 6th root as a \(1/6\) power.

\begin{equation*} \ln\left(x^{13}\right)-\ln\left(z^7\right)-\ln\left((x+1)^{1/6}\right) \end{equation*}

Now we can get our final answer by bringing down all the exponents.

\begin{equation*} 13\ln\left(x\right)-7\ln\left(z\right)-\frac{1}{6}\ln\left(x+1\right) \end{equation*}