Given that log_2(x)=p and log_4(y)=q, how do I evaluate log_x(4y)?There were some other questions like this and I applied this formula to them log_a(xy) = log_a(x)+log_a(y).However, in this question, I can't because I can't evaluate log_x(y). What do I do?

Landen Miller

Landen Miller

Open question

2022-08-18

Given that log 2 ( x ) = p and log 4 ( y ) = q, how do I evaluate log x ( 4 y )?
There were some other questions like this and I applied this formula to them log a ( x y ) = log a ( x ) + log a ( y ). However, in this question, I can't because I can't evaluate log x ( y ). What do I do?

Answer & Explanation

Jason Petersen

Jason Petersen

Beginner2022-08-19Added 13 answers

log x ( 4 y ) = ln 4 + ln y ln x = ln 4 + q ln 4 p ln 2 = 2 ln 2 ( q + 1 ) p ln 2 = 2 ( q + 1 ) p
Dorsheele0p

Dorsheele0p

Beginner2022-08-20Added 5 answers

log x 4 y = log x 4 + log x y
We know that log x a = log 4 a log x , so log x 4 y = log 4 4 + log 4 y log 4 x = q + 1 log 4 x
We also know log 2 x = p, so 2 p = x
and
( 4 1 2 ) p = x
and
4 p 2 = x
So log 4 x = p 2 , and we're done.
log x 4 y = 2 ( q + 1 ) p

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