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.

m = n \Leftrightarrow ln (m) = ln (n)

Note that in order to take logarithms, both

m

and

n

must be positive numbers. Let's now solve our equation.

2 e^{2 x} - 7 = 5

AddEqn

LHS + 7 = RHS + 7

2 e^{2 x} = 12

DivEqn

LHS / 2 = RHS / 2

e^{2 x} = 6

ln (LHS) = ln (RHS)

ln (e^{2 x}) = ln (6)

ln (e^{a}) = a

2 x = ln (6)

Calculate logarithm

2 x = 1.791759469 \dots

DivEqn

LHS / 2 = RHS / 2

x = 0.8958797346 \dots

RoundDecRound to

x \approx 0.896

The Natural Base e 1.12 - Solution