How do I find the inverse of a log function?

[hancyu]

Homework Statement

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?

Homework 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)

The Attempt at a Solution

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

 

[hancyu]

also, is the inverse of

f(x) = 2^x−1 + 3

log₂ (x/3) +1 = y


f(x) = log_2/3(x − 2) − 4

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

are these correct?

 

[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), that's 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 =]

 

[HallsofIvy]

Presumably you know that the domain and range of any function of the form log_a(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 log_a(f(x))/log_a(g(x)) is all x such that x is positive and g(x) is not 1 (so that log(g(x)) is not 0).

 

[hancyu]

Presumably you know that the domain and range of any function of the form log_a(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 log_a(f(x))/log_a(g(x)) is all x such that x is positive and g(x) is not 1 (so that log(g(x)) is not 0).

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

i still can't get the inverse tho...