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The Natural Base e

The Natural Base e 1.4 - Solution

When bases are not the same, we can solve an exponential equation by taking logarithms on each side of the equation. In our case, since we have the number $e$ as a base, we will take the natural logarithm of each side. $\begin{gathered} m=n \quad \Leftrightarrow \quad \ln (m) = \ln (n) \end{gathered}$ Note that in order to take logarithms, both $m$ and $n$ must be positive numbers. Let's now solve our equation.
$3e^{4x}+9=15$
$3e^{4x}=6$
$e^{4x}=2$
$\ln(\text{LHS})=\ln(\text{RHS})$
$\ln e^{4x}= \ln (2)$
$\ln\left(e^a\right) = a$
$4x= \ln (2)$
Calculate logarithm
$4x=0.6931471806\ldots$
$x=0.1732867951\ldots$
$x\approx 0.173$