Question

I have an ACT practice problem that I’m having trouble on

Answer 1

Given below the logarithm terms

$\log _7x+\log _7y-\log _7z$

In order to resolve the above problem, we will apply two properties of the logarithm.

These properties are,

$\begin{gathered} \log _c\mleft(a\mright)+\log _c\mleft(b\mright)=\log _c\mleft(ab\mright)\ldots\ldots.1 \\ \quad \log _c\mleft(a\mright)-\log _c\mleft(b\mright)=\log _c\mleft(\frac{a}{b}\mright)\ldots\ldots.2 \end{gathered}$

Hence,

$\begin{gathered} \log _7x+\log _7y-\log _7z=\log _7\frac{\mleft(x\times y\mright)}{z} \\ \log _7x+\log _7y-\log _7z=\log _7\mleft(\frac{xy}{z}\mright) \end{gathered}$

Therefore,

$\log _7x+\log _7y-\log _7z=\log _7(\frac{xy}{z})$

The error he did was that instead of multiplying, he added, and also instead of dividing he subtracted.

The correct answer is,

$\log _7(\frac{xy}{z})$