Solve 6^2x+1=1296

$${ 6 }^{ 2x+1 } =1296$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Definition of Logarithm

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

$$6^{2x+1}=1296$$

Take the logarithm of both sides of the equation.

$$\log(6^{2x+1})=\log(1296)$$

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

$$\left(2x+1\right)\log(6)=\log(1296)$$

Divide both sides by \log(6).

$$2x+1=\frac{\log(1296)}{\log(6)}$$

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

$$2x+1=\log_{6}\left(1296\right)$$

Subtract 1 from both sides of the equation.

$$2x=4-1$$

Divide both sides by 2.

$$x=\frac{3}{2}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(6)}+\frac{3}{2}$$

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