Hi everyone

I am trying to solve this question

Prove that

LOG_{4}X=LOG_{2}√X

Thanks in advance

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- Dec 17th 2012, 07:39 AMlostinlalalandProve logarithmic formula is accurate
Hi everyone

I am trying to solve this question

Prove that

LOG_{4}X=LOG_{2}√X

Thanks in advance - Dec 17th 2012, 07:48 AMMarkFLRe: Prove logarithmic formula is accurate
Hint: prove the identity:

$\displaystyle \log_{a^n}(b)=\frac{\log_a(b)}{n}$

with the change of base formula:

$\displaystyle \log_a(c)=\frac{\log_b(c)}{\log_b(a)}$ - Dec 17th 2012, 07:57 AMlostinlalalandRe: Prove logarithmic formula is accurate
Thanks for the quick reply.

What I understood is that I should treat it as change base of log4x to base 2

Though I am sure you are pointing me in the correct direction my brain is dead after looking at this for the past 2 hours - Dec 17th 2012, 08:00 AMMarkFLRe: Prove logarithmic formula is accurate
Yes, a more specific way to go would be to write:

$\displaystyle \log_4(a)=\frac{\log_2(a)}{\log_2(4)}=\frac{\log_2 (a)}{\log_2(2^2)}=$... - Dec 17th 2012, 08:08 AMlostinlalalandRe: Prove logarithmic formula is accurate
Thanks, nice car by the way

What confuses me is the square root if I follow through the equation I feel I will still end up with an incorrect fraction - Dec 17th 2012, 01:48 PMDevenoRe: Prove logarithmic formula is accurate
it is a rule of logarithms that:

log_{n}(a^{b}) = b(log_{n}(a)), even when b = 1/2..... - Dec 17th 2012, 02:10 PMskeeterRe: Prove logarithmic formula is accurate
- Dec 19th 2012, 12:43 AMibduttRe: Prove logarithmic formula is accurate
See the image for solution

- Dec 19th 2012, 01:06 AMBobPRe: Prove logarithmic formula is accurate
Why bother with change of base ? You don't need it, (and I always struggle to remember it anyway, I finish up having to look it up to be sure I'm remembering it correctly).

Let $\displaystyle \log_{4}X=m, \text{ then } X=4^{m}............(1)$

Let $\displaystyle \log_{2}\sqrt{X}=n, \text{ then } \sqrt{X}=2^{n} \Rightarrow X=4^{n}............(2)$

Comparison of (1) and (2) shows that $\displaystyle m=n.$