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I have a problem:

Let $f(x)=\ln(x)$ solve each of the following equations for $x$.

the question is in three parts

  1. $(f(x))^{-1}=5$

  2. $f^{-1}(x)=5$

  3. $f(x^{-1})=5$

My understanding is that $\ln(x)$ is the same as $\log_e X=\text{exponent}$

So in item $2$ the answer should be $e^5=X$right? which is $148.413\ldots$, but this is wrong. What am I missing?

Michael Hardy
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codenamejupiterx
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    It is not clear when you mean the inverse function (which would be the exponential function) and when you mean the reciprocal. – marty cohen Feb 12 '13 at 01:50

2 Answers2

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Let $f(x)=\ln(x)$ solve each of the following equations for $x$.

$$f(x)=\ln x$$

$$f^{-1}(x)=e^x$$

$$(f(x))^{-1}=e^x$$

$$f(x^{-1})=\ln(x^{-1})$$

a. $$(f(x))^{-1}=5$$

$$(\ln(x))^{-1}=5$$ $$\ln^{-1}(x)=5$$ $$e^x=5$$ $$\ln(e^x)=\ln(5)$$ $$\ln(5)=x$$

b. $$f^{-1}(x)=5$$

$$e^x=5$$ $$\ln(e^x)=\ln(5)$$ $$x=\ln(5)$$

c. $$f(x^{-1})=5$$ $$f(x^{-1})=\ln(x^{-1})$$ $$\ln(x^{-1})=5$$ $$(-1)\ln(x)=5$$ $$\ln(x)=-5$$ $$e^{\ln(x)} = e^{-5}$$ $$x=\frac{1}{e^{5}}$$

Community
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user42538
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$$ \ln^{-1} x = 5 $$ is the same as $$ e^x = 5 $$ or the same as $$ x = \ln 5. $$

(Please don't write $X$ if you mean $x$. Mathematical notation is case-sensitive.)

Michael Hardy
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