Question

This is one practice problem that I need help on It asks to solve logarithmic equations, and It includes a question within it. Please note that this is a lengthy problem. & I have enlarged the equations for better view.

Answer 1

The Solution:

We are required to solve each of the following logarithmic equations:

a.

$\begin{gathered} 1.\text{ }\log _464=m \\ \text{cross multiplying, we get} \\ 64=4^m \\ 4^3=4^m \\ m=3 \end{gathered}$$\begin{gathered} 2.\text{ }\log _8n=3 \\ \text{Cross multiplying, we get} \\ n=8^3=512 \end{gathered}$$\begin{gathered} 3.\text{ }\log _p4096=3 \\ \text{Cross multiplying, we get} \\ 4096=p^3 \\ \text{ Equalising the base in both sides, we get} \\ 16^3=p^3 \\ 16=p \\ p=16 \end{gathered}$

$\begin{gathered} 4.\text{ }\log _{32}q=3 \\ \text{cross multiplying, we get} \\ q=32^3=32\times32\times32=32768 \end{gathered}$

b.

Given the equation below:

$\log _{4^x}2^a=3$

We are required to express a in terms of x.

$\begin{gathered} \log _{4^x}2^a=3 \\ \text{Cross multiplying, we get} \\ 2^a=(4^x)^3 \\ \text{Equalizing the base, we get} \\ 2^a=(2^{2x})^3 \\ 2^a=2^{6x} \\ a=6x \end{gathered}$