You are proving that the Power Rule of Logarithms holds true for logarithms of any positive base that is not equal to 1. The following is your work so far.

Question

Let logbm=x
log
𝑏
⁡
𝑚
=
𝑥
. This is equivalent to bx=m
𝑏
𝑥
=
𝑚
, which is the same as m=bx
𝑚
=
𝑏
𝑥
. Therefore, mn=(bx)n
𝑚
𝑛
=
(
𝑏
𝑥
)
𝑛
.

Which of the following options best describes the next step you should take in this proof?

(1 point)
Responses

Substitute specific values for m, n, and x.
Substitute specific values for m , n , and x .

Rewrite this exponential equation as a logarithmic equation.
Rewrite this exponential equation as a logarithmic equation.

Substitute logbm
log
𝑏
⁡
𝑚
for x.
Substitute logarithm subscript b baseline m for x.

Simplify this equation using the Power Rule of Exponents.

Answer 1

To continue the proof of the Power Rule of Logarithms, the best step to take next is:

Simplify this equation using the Power Rule of Exponents.

This step involves manipulating the exponential equation $mn=(b^x{)}^n$, which can then be simplified to $m^n=b^{nx}$. From there, you can apply logarithmic definitions and properties to ultimately prove the Power Rule for logarithms, which states that ${\log }_b(m^n)=n\cdot {\log }_b(m)$.