Solve 1/5e^6x=4

$$\frac { 1 } { 5 } e ^ { 6 x } = 4$$

Solve for x

$$x=\frac{\ln(20)}{6}\approx 0.499288712$$

Steps Using Definition of Logarithm

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

$$\frac{1}{5}e^{6x}=4$$

Multiply both sides by 5.

$$e^{6x}=20$$

Take the logarithm of both sides of the equation.

$$\log(e^{6x})=\log(20)$$

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

$$6x\log(e)=\log(20)$$

Divide both sides by \log(e).

$$6x=\frac{\log(20)}{\log(e)}$$

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

$$6x=\log_{e}\left(20\right)$$

Divide both sides by 6.

$$x=\frac{\ln(20)}{6}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{3}+\frac{\ln(20)}{6}$$

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