How do we prove that a <mrow class="MJX-TeXAtom-ORD"> log &#x2061;<!--

Jerry Villegas

Jerry Villegas

Answered question

2022-05-26

How do we prove that a log  b = b log  a for a > 1 and b > 1?
Although I've tried, I can't quite complete the equality using the change of base formula:
a log  b  a log a  b log a  a 
How can I make the exponent's base be b?

Answer & Explanation

Leah Conley

Leah Conley

Beginner2022-05-27Added 12 answers

Since x = e log x
a log b = ( e log a ) log b = e log a log b = e log b log a = ( e log b ) log a = b log a
Carly Roy

Carly Roy

Beginner2022-05-28Added 4 answers

Let (capital) B be the base of the logarithms. Then
a log B  b = ( B log B  a ) log B  b = B ( log B  a ) ( log B  b ) ,
and that is clearly symmetric in a and b. You may also simply continue from there:
 = ( B log B  b ) log B  a = b log B  a .

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