A little help here with simplifying logarithmic equations

[bjuveges]

[h=2]Problem 12[/h]For what value of does


[h=2]Solution[/h]




OK, I understand completely how to solve this problem, and by that I mean the process that is required to find the answer. But what I am having trouble with is the simplification of logs in this equation. I dont understand what happened in this solution to where every set = log(2)(x)

Like for example: How did they simplify (3log(2)(x))/(log(2)(8)) and get log(2)(x)?

except apply this for every set of logs that were simplified, I must be missing something because I thought I understood logarithmic equations well.

Anyways, any help is appreciated

Thanks

 

[Quaid]

Hi:

Here's some stuff that we ought to memorize, before working with logarithms.

sqrt(n) = n^(1/2)

For example, sqrt(2) = 2^(1/2)

First few powers of 2:

2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64

Therefore, since logs are exponents, we have

log₂(4) = 2
log₂(8) = 3
log₂(16) = 4
log₂(32) = 5
log₂(64) = 6

et cetera.

Also, a very important property of logarithms:

n*log(x) = log(x^n)

Use it like so,

\(\displaystyle 2 \cdot log_2(\sqrt{2})\)

\(\displaystyle 2 \cdot log_2(2^{½})\)

\(\displaystyle log_2(2^{2(½)})\)

\(\displaystyle log_2(2)\)

Cheers