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most of the time I know how to find the inverse of a function (make it equal $y$, solve for $x$ and then swap $x$ and $y$), but I have no idea how to do that for this one, so any help would be great: $h(x)=3^x$

From the question:

Solve the equation $h^{-1}(x)=2$

Thanks in advance for any help!

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    As you can see from Simon's comment, the question isn't actually asking you to find the inverse function $h^{-1}$. If that's what you needed to do, you would indeed need logarithms, but as it is, you can treat $h^{-1}$ as a black box without worrying about what function it is, and just write $x=h(h^{-1}(x))=h(2)=3^2=9$.2012-05-02

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I think both existing answers are sort of missing the point of the exercise. You don't need to know anything about logarithms to do this exercise; all you need are the formal properties of inverse functions. The solution

$x=h(h^{-1}(x))=h(2)=3^2=9$

uses only the specific form of the function $h$ and the general formal properties of inverse functions, not the specific form of $h^{-1}$.

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    That's a great answer! Thanks.2012-05-02
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You know the process of finding the inverse, so let's go through it step by step.

First, replace $h(x)$ with $y$, $\quad y = 3^x$. Then, switch the $x$ and the $y$, $\quad x = 3^y$. Now, you have to solve for $y$ to find the inverse function. We can't take the $y$-th root of both sides, so in order to solve for $y$, we want to find the exponent that turns 3 into $x$. This is what's called a logarithm. By definition. $y = \log_ax$ if and only if $x = a^y$, that is, $\log_ax$ is the exponent that turns the base $a$ into $x$.

With this in mind, the inverse of $h(x) = 3^x$ would be $h^{-1}(x) = \log_3x$.

To solve the equation $h^{-1}(x) = 2$ we use the above definition.

$ h^{-1}(x) = 2 \rightarrow \log_3x = 2 \rightarrow x = 3^2 \rightarrow x = 9.$

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(I'd like to comment but I don't have that much reputation, sorry...)

You do need logarithms to invert this exponential function! The inverse will be

$ h^{-1}(x)=\log_3x $

Therefore, let $2=\log_3x$, we immediately see that $x=3^2=9$.

Additionally: Thanks to @joriki, I believe I need to make things clear. This exercise requires you to establish a better understanding of what an inverse to a function actually means. See joriki's answer!

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    @DaweiWang: I see, sorry.2012-05-02
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  1. First replace $h(x)$ with $y$ to get $y=3^x$.
  2. Swap $x, y$ to get $x=3^y$.
  3. Take, for example, $\log_3$ of both sides to get $\log_3x=y\log_33=y.$
  4. Replace $y$ with $h^{-1}(x)$.
  5. $h^{-1}(x)=\log_3(x)$.
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Denoting f{x) =y, just interchange x and y . It is that simple and also it is done!

$y=3^x $ becomes

$x=3^y , $ or $y_{inv} = \log_3(x) $

Now if $ \log_3(x) =2, x= 3^2 =9. $

You can graph $ f(x) $ and $ f^{-1}(x) $ to check that they indeed intersect on

the line $x=y,$ and nothing more needs to be done.

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The best way to solve the question is $h^{-1}(x) =2$ then $x=h (2)$

Now solve $h (2) =3^2=9$

So $x=9$

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    Please use [MathJax](http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-qu%E2%80%8C%E2%80%8Bick-reference) to type your answer. By the way, I don't understand why you are answering this answerd question !!2015-12-15