# Inverse of log function

1. Sep 30, 2008

### hancyu

1. The problem statement, all variables and given/known data

f(x) = log2 x + 3
2 log2 x − 1
how do i find the inverse of this? how do i find the range and domain of a log function?

2. Relevant equations

f(x) = log2 x + 3
2 log2 x − 1

is equal to
f(x) = log2 x + 3 - log2 (x − 1)2

D of f(x) = R of f-1(x)

3. The attempt at a solution

i tried changing the base but it didnt work...

Last edited: Oct 1, 2008
2. Sep 30, 2008

### hancyu

also, is the inverse of

f(x) = 2x−1 + 3

log2 (x/3) +1 = y

f(x) = log2/3(x − 2) − 4

(2/3)x+4 + 2 = y

are these correct?

3. Oct 1, 2008

### Gib Z

Ok. Your attempt was sadly, incorrect, although similar to something you were probably thinking of:

$$\log_c ( \frac{a}{b} ) = \log_c a - \log_c b$$,

which is not the same as what you tried: $$\frac{ \log_c a}{\log_c b} = \log_c a - \log_c b$$, which is not true.

It might help if you let $u= log_2 x$ so that you may view the problem easier. Doing so, solve the equation you have for you, replace the expression in x back in and solve it for x. Then swap your x and f(x), thats your inverse function!

For your second problem, not quite. Solve it for x first. So First take 3 to the other side,

$2^{x-1} = f(x) - 3$. After that, take log_2 of both sides, hopefully you can see the rest. Then just swap x for f(x).

The last one looks correct, good work =]

4. Oct 1, 2008

### HallsofIvy

Staff Emeritus
Presumably you know that the domain and range of any function of the form loga(x) is {x|x> 0} and all real numbers respectively.

You also should know that the domain of a rational function is all numbers such that the denominator is not zero.

Putting those together, the domain of loga(f(x))/loga(g(x)) is all x such that x is positive and g(x) is not 1 (so that log(g(x)) is not 0).

5. Oct 1, 2008

### hancyu

so the domain of the 1st one is x=>0 ? because log can never be zero or negative?

i still cant get the inverse tho...