The Natural Base e

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

SubEqn

\text{LHS}-9=\text{RHS}-9

3e^{4x}=6

DivEqn

\left.\text{LHS}\middle/3\right.=\left.\text{RHS}\middle/3\right.

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

DivEqn

\left.\text{LHS}\middle/4\right.=\left.\text{RHS}\middle/4\right.

x=0.1732867951\ldots

RoundDecRound to

x\approx 0.173

The Natural Base e 1.4 - Solution