Solve 11^2x+1=121

$$11 ^ { 2 x + 1 } = 121$$

Solve for x

$$x=\frac{1}{2}=0.5$$

Steps Using Definition of Logarithm

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

$$11^{2x+1}=121$$

Take the logarithm of both sides of the equation.

$$\log(11^{2x+1})=\log(121)$$

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

$$\left(2x+1\right)\log(11)=\log(121)$$

Divide both sides by \log(11).

$$2x+1=\frac{\log(121)}{\log(11)}$$

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

$$2x+1=\log_{11}\left(121\right)$$

Subtract 1 from both sides of the equation.

$$2x=2-1$$

Divide both sides by 2.

$$x=\frac{1}{2}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(11)}+\frac{1}{2}$$

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