Solve 5*e^3a=90

$$5 \cdot e ^ { 3 a } = 90$$

Solve for a

$$a=\frac{\ln(18)}{3}\approx 0.963457253$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$5e^{3a}=90$$

Divide both sides by 5.

$$e^{3a}=18$$

Take the logarithm of both sides of the equation.

$$\log(e^{3a})=\log(18)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$3a\log(e)=\log(18)$$

Divide both sides by \log(e).

$$3a=\frac{\log(18)}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$3a=\log_{e}\left(18\right)$$

Divide both sides by 3.

$$a=\frac{\ln(18)}{3}$$

Solve for a (complex solution)

$$a=\frac{2\pi n_{1}i}{3}+\frac{\ln(18)}{3}$$

$$n_{1}\in \mathrm{Z}$$