Question

Use like bases and the one-to-one property to solve each exponential equation. using logarithms

Answer 1

The given equation is:

$64\times4^{3x}=16$

It is required to solve the equation using logarithms.

Divide both sides of the equation by 64:

$\begin{gathered} \frac{64\times4^{3x}}{64}=\frac{16}{64} \\ \Rightarrow4^{3x}=\frac{1}{4} \\ \text{ Take the logarithm to base }4\text{ of both sides:} \\ \Rightarrow\log_44^{3x}=\log_4\frac{1}{4} \\ \Rightarrow\log_44^{3x}=\log_44^{-1} \\ \end{gathered}$

Using like bases and the one-to-one property, it follows that the exponents must be equal:

$\begin{gathered} 3x=-1 \\ \text{ Divide both sides by }3: \\ \Rightarrow\frac{3x}{3}=\frac{-1}{3} \\ \\ \Rightarrow x=-\frac{1}{3} \end{gathered}$

The answer is x=-1/3.