Confusion regarding the Logarithmic function change of base formula My textbook seems to be making

arridsd9

arridsd9

Answered question

2022-06-21

Confusion regarding the Logarithmic function change of base formula
My textbook seems to be making a big leap when trying to prove the change of base formula for logarithms. If someone could help clear this up it would be very appreciated.
It starts with:
b x log b ( a )
and uses the power rule to get:
b x log b ( a ) = b log b ( a x )
And it equates all this to:
b x log b ( a ) = b log b ( a x ) = a x
Okay, I get it up to here, but then for me it leaps from that to this:
log a ( x ) log b ( a ) = log b ( a log a ( x ) ) = log b ( x )
And it says that divide through by log b ( a ) to get the result.
What precisely has happened here? Could someone walk me through this step-by-step?
Thank you.

Answer & Explanation

massetereqe

massetereqe

Beginner2022-06-22Added 21 answers

I would emphasize these sorts of things:
b log b t = t
and
log b ( b t ) = t .
Your third line says
b x log b a = a x .
I would take log b of both sides to get the one i actually remember
log b ( a x ) = x log b a .
That is, to take log with an exponent, pull the exponent in front and apply the log to what remains.
To match up (eventually) with your text, let me replace theletter x with a letter t, for
log b ( a t ) = t log b a .
Now, make the purely algebraic substitution
t = log a x .
log b ( a log a x ) = log a x log b a .
However,
a log a x = x .
So
log b ( x ) = log a x log b a .
Or
log a x = log b x log b a .
Notice how the b's seem to cancel in the fraction on the right hand side. Something to help remember.
For example, with a = 4 , b = 2 , x = 64 , we see
3 = log 4 64 = log 2 64 log 2 4 = 6 2 = 3
Emmy Dillon

Emmy Dillon

Beginner2022-06-23Added 5 answers

a L o g ( x ) = L o g ( x a ) according to the rules of logarithms. Also, a L o g a x = x. I think that this gets you there.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?