Question

Find the solution to the equation 64^(3 – x) = 4^2x. Using complete sentences, explain the procedure used to solve this equation.

Answer 1

Let's start by stating that:
\[64=4^{3}\]
Now look at the problem:
\[64^{3-x}=4^{2x}\]
Substitute in the first statement:
\[4^{3(3-x)}=4^{2x}\]
\[4^{9-3x}=4^{2x}\]
Now we take the log base 4 on each side to remove the 4.
\[\log_{4} 4^{9-3x}=\log_{4} 4^{2x}\]
Taking the log base 4 cancels the 4 on each side so we get:
\[9-3x=2x\]
Now we simplify:
\[9=5x\]
\[x=\frac{9}{5}\]