Solve 6^2x-1=1/6

$$6 ^ { 2 x - 1 } = \frac { 1 } { 6 }$$

Solve for x

$$x=0$$

Steps Using Definition of Logarithm

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

$$6^{2x-1}=\frac{1}{6}$$

Take the logarithm of both sides of the equation.

$$\log(6^{2x-1})=\log(\frac{1}{6})$$

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

$$\left(2x-1\right)\log(6)=\log(\frac{1}{6})$$

Divide both sides by \log(6).

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

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

$$2x-1=\log_{6}\left(\frac{1}{6}\right)$$

Add 1 to both sides of the equation.

$$2x=-1-\left(-1\right)$$

Divide both sides by 2.

$$x=\frac{0}{2}$$

Solve for x (complex solution)

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

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