Inverse of natural log

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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

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?

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2 Answers

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

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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.)

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