Patrick T. answered • 11/24/21

Tutor Specializing in French & Math (up to college Pre-Calculus)

Hello Byanca,

Assuming the question indeed asks to simplify log_{d}(m) / log_{d}(2):

You want to use the change-of-base formula for logs, but I'll use the definition of log_{a}(x) to break down the question.

**log**_{a}**(x) = log(x)/log(a) --- #1**

Now using it for your question:

log_{d}(m) = log(m)/log(d)

log_{d}(2) = log(2)/log(d)

Therefore: log_{d}(m) / log_{d}(2) = [log(m)/log(d)] / [log(2)/log(d)]

Division by a fraction means you multiply by the reciprocal of the bottom fraction so:

log_{d}(m) / log_{d}(2) = [log(m)/log(d)] * [log(d)/log(2)]

You can cross out log (d) which leaves you with:

log_{d}(m) / log_{d}(2) = [log(m)/log(2)]

Going back to equation #1, log(m)/log(2) can be expressed as **log**_{2}**(m)**

Cheers.