Question

How do you solve $\ln (5{{e}^{6}})$ ?

Answer 1

Hint: The function $y={{e}^{x}}$and $y=\ln x$are inverse functions. To solve this problem, we need to know the logarithmic and radical property which is $\ln {{a}^{b}}=b\ln a$ and $\ln (ab)=\ln a+\ln b$.

Complete Step By Step solution:
According to the given information, we have to solve $\ln (5{{e}^{6}})$.
We know that the natural log ($\ln $) and the exponential function (${{e}^{x}}$) are inverses of each other which simply means that they will cancel out each other resulting in $1$.
Also, by using the property $\ln (ab)=\ln a+\ln b$, we can write
$\ln (5{{e}^{6}})=\ln 5+\ln {{e}^{6}}$
Using the property $\ln {{a}^{b}}=b\ln a$ we can write $\ln {{e}^{6}}$ as,
$6\ln e$
In this case if I know that as $\ln $ and $e$ are inverse of each other, they will simply undo each other.
$\Rightarrow \ln (5{{e}^{6}})=\ln 5+6\ln e$
$\Rightarrow \ln (5{{e}^{6}})=\ln 5+6$

Hence the result will be $=\ln 5+6$.

Note:
We know that the natural log ($\ln $) and the exponential function (${{e}^{x}}$) are inverses of each other, which means that if we raise the exponential function by $\ln $ of $x$, we would be able to find $x$. But remember, if something is done on one side of the equation, the same thing has to happen on the opposite side of the equation too.