### Home > INT3 > Chapter 10 > Lesson 10.1.5 > Problem10-72

10-72.

Write three different, but equivalent, expressions for each of the following log expressions. For example: $\text{log}\left(7^{3/2}\right)$ can be written as $\frac{3}{2}\log\left(7\right),\frac{1}{2}\log\left(7^3\right),3\log(\sqrt{7})$, etc.

Use the properties of logarithms and exponents to help you rewrite these.
Simplify different parts of the expression to create new, equivalent expressions.

1. $\text{log}\left(8^{2/3}\right)$

$\text{The Power Property of Logarithms is log}_m(a^n)=n\cdot \log m(a), \text{ so one way to rewrite this expression is }\frac{2}{3}\log(8).$

2. $–2\text{log}\left(5\right)$

Use the Power Property of Logarithms in reverse here.
Since $n · \text{log}_{m}\left(a\right) = \text{log}_{m}\left(a^{n}\right), −2 \text{log}\left(5\right) = \text{log}\left(5^{−2}\right)$.

Find two different ways to rewrite $5^{−2}$
(there are more than two ways to do this).

3. $\text{log}\left(na\right)^{bo}$

Refer to parts (a) and (b).

$o\text{log}\left(n^{b}a^{b}\right), b\text{log}\left(na\right)^{o}, bo\text{log}\left(na\right)$