Solve 5(e^2x-10)=115

$$5 ( e ^ { 2 x } - 10 ) = 115$$

Solve for x

$$x=\frac{\ln(33)}{2}\approx 1.748253781$$

Steps Using Definition of Logarithm

Divide both sides by 5.

$$e^{2x}-10=\frac{115}{5}$$

Divide 115 by 5 to get 23.

$$e^{2x}-10=23$$

Add 10 to both sides of the equation.

$$e^{2x}=33$$

Take the logarithm of both sides of the equation.

$$\log(e^{2x})=\log(33)$$

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

$$2x\log(e)=\log(33)$$

Divide both sides by \log(e).

$$2x=\frac{\log(33)}{\log(e)}$$

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

$$2x=\log_{e}\left(33\right)$$

Divide both sides by 2.

$$x=\frac{\ln(33)}{2}$$

Solve for x (complex solution)

$$x=\frac{\ln(33)}{2}+\pi n_{1}i$$

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