Changing base of a logarithm by taking a square root from base? From my homework I found

crossoverman9b

crossoverman9b

Answered question

2022-06-20

Changing base of a logarithm by taking a square root from base?
From my homework I found
log 9 x = log 3 x
and besides that an explanation that to this was done by taking a square root of the base. I fail to grasp this completely. Should I need to turn log 9 x into base 3, I'd do something like
log 9 x = log 3 x log 3 9 = log 3 x log 3 3 2 = log 3 x 2
but this is a far cry from what I've given as being the correct answer.
Substituting some values to x and playing with my calculator I can see that the answer given as correct is correct whereas my attempt fails to yield the correct answer.
Now the question is, what are correct steps to derive log 3 x from log 9 x ? How and why am I allowed to take a square root of the base and the exponent?

Answer & Explanation

mallol3i

mallol3i

Beginner2022-06-21Added 20 answers

You are only one step away!
Note that a log b = log b a , so
log 3 x 2 = 1 2 log 3 x = log 3 x 1 / 2 = log 3 x .
Yesenia Sherman

Yesenia Sherman

Beginner2022-06-22Added 5 answers

log 9 x = a 9 a = x log 3 x = b 3 b = x ( 3 b ) 2 = ( x ) 2 3 2 b = x 9 b = x   } a = b

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