Solve e^9x-13=76

$${ e }^{ 9x-13 } =76$$

Solve for x

$$x = \frac{\log_{e} {(76)} + 13}{9} \approx 1.925637038$$

Steps Using Definition of Logarithm

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

$$e^{9x-13}=76$$

Take the logarithm of both sides of the equation.

$$\log(e^{9x-13})=\log(76)$$

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

$$\left(9x-13\right)\log(e)=\log(76)$$

Divide both sides by \log(e).

$$9x-13=\frac{\log(76)}{\log(e)}$$

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

$$9x-13=\log_{e}\left(76\right)$$

Add 13 to both sides of the equation.

$$9x=\ln(76)-\left(-13\right)$$

Divide both sides by 9.

$$x=\frac{\ln(76)+13}{9}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{9}+\frac{\ln(76)+13}{9}$$

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