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What kind of rule or formula this kind of equations uses?

For example we have:

$$a=e^{x}$$

How come it is equal to:

$$\ln a =x$$

Tried to find some kind of rule for that about how it works, but didn't found anything.

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    Are you familiar with the *inverse* of a function, as a general concept? This is a particular case of a function's inverse. – 2017-02-04
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    it's its inverse! – 2017-02-04
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    Instead of using the Google API to produce an *image* of an equation, you can [post mathematical expressions](http://math.stackexchange.com/help/notation) here using $\LaTeX$ syntax. – 2017-02-04
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    Okey thanks for info @hardmath – 2017-02-04
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    And oh yes, right, it is the inverse – 2017-02-04

2 Answers 2

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Note $\log_a x$ is the inverse of the function $a^x$. When we speak of the natural log of $x$, it is written $\ln(x)$, which is simply shorthand for $\log_e(x)$.

$$a = e^x$$

Since $\log x$, or in this case the natural log $\log_e x = \ln x$ is a strictly increasing function, we can take the $\ln$ of both sides to get: $$\ln(a) = \ln(e^x) = x\ln(e) = x$$

Note that for any real $a, b$, be have $\underbrace{\log_a(a^b) = b\log_a(a)}_{\log x^y = y \log x} = b\cdot 1 = b$.

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    You're welcome. RedRose. – 2017-02-04
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    Btw how come b*1=1? – 2017-02-04
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    Because $\log_a(a) = 1,$ just like $\log_2(2) = 1$, just as $\ln(e) = 1$. If we raise each side of $\log_e(e) = \ln(e)= 1$, as a power a power of e, we have $e^{\ln e} = e^1 \iff \ln e = 1$ – 2017-02-04
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    Yes right, I understood that log_a(a) = 1, but shouldn't b*1=b? – 2017-02-04
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    Yikes! Major typo, yes you're spot on. Corrected. – 2017-02-04
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    Okey, thanks :) – 2017-02-04
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The rule we have is the definition of logarithim.The logarithim, by definition is the inverse of the exponential function.

Note that by the definition of logarithim $$a=b^{x}$$ becomes $$\log_{b} a=x$$ Yours is just a case when $b=e$. Note $\log_{e} x =\ln x$.If you're curious about $\log$ and $\ln$, see more here.