How do i solve this logarithmic expression?!-10 points (maths)?

Question

If log4(x)=p, show that log2(x) =2p, well this is fairly simple and ive accomplished it. The problem is the next part.

Hence, find a) the value of k if log4(k) = 2+ log2(k) and b) the value of n if log2(n) + log4(n) =9

Well this is what i did==> log4(k) -log2(k) = 16====> log4(k)/ log2(k) =16 ....what now! Please help withh method and explanation!

Answer both questions for 10 points!

Answer 1

Favorite Answer

Good job on your representation of log(base), i.e. log4(x), you may also see the following notations:

log_4(x) = p , log_2(x) = 2p, or log₄(x) = p , log₂(x) = 2p

solution: change the bases to either 10 or e(natural log):

log₂(x) = log(x)/log(2)

log₄(x) = log(x)/log(4) = p

log(x)/log(2²) = p

log(x)/2log(2) = p

log(x)/log(2) = 2p = log₂(x)

a)

log₄(k) = 2 + log₂(k)

log(k)/log(4) - log(k)/log(2) = 2

[log(k) - 2log(k)]/ [2log(2)] = 2

-log(k) = 2 * 2log(2)

log(2^4) + log(k) = 0

log(16k) = 0

16k = 10^0 = 1

k = 1/16

All the best.

Answer 2

Let log k to the base 2 = a.

So k = 2^a

Let log k to the base 4 = b

So k = 4^b = (2^2)^b = 2^(2b)

Therefore 2^a = 2^(2b)

and so a = 2b

Given that b = 2 + a

Hence a = 2(2 + a), a = -4, b = -2, k = 4^(-2) = 1 / 4^2 = 1/16