Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have a homework problem that I am struggling to understand. The problem is Find a formula for the inverse function $f^ {-1}$ of the function $f$. $$f(x)=\log_{2x}3$$

Here is my attempt at solving this problem. $$y=\log_{2x}3$$ $$2x ^y=3$$ $$x^y=3/2$$ $$(x ^y)^{1/y}=\left(\frac{3}{2}\right)^{1/y}$$ $$x=\left(\frac{3}{2}\right)^{1/y}$$

However, the answer in the book is $\frac{3^{1/y}}{2}$. I think I know my mistake but I am not sure why it is wrong. I believe that my mistake was when I divided $2$ from $2x^y$, are you not allowed to do that?

share|improve this question
add comment

2 Answers 2

up vote 2 down vote accepted

The mistake is in the first line. You should have $(2x)^y$ instead of $2(x^y)$. $(2x)$ is the base of the logarithm and so the entire term needs to be taken to the power of $y$.

share|improve this answer
    
Since the entire term is is raised to the power of $y$, I would have to take care of the $y$ first before dividing by 2 correct? I think I understand now. –  Kot Oct 4 '12 at 23:36
    
Exactly! You don't have $2$ alone in the equation but rather a factor of $2^y$. You not only need to take care of the $y$ atop the $x$ but also the $2$. –  EuYu Oct 4 '12 at 23:40
add comment

Well, you almost got it). If $y=f(x)$ then $x=f^{-1}(y)$.

$$ y=f(x)=\log_{2x}3\\ (2x)^y=3\\ 2x=3^{\frac{1}{y}}\\ x=\frac{1}{2}3^{\frac{1}{y}}=f^{-1}(y) $$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.