Solve 6^5x-7=36

$$6 ^ { 5 x - 7 } = 36$$

Solve for x

$$x = \frac{9}{5} = 1\frac{4}{5} = 1.8$$

Steps Using Definition of Logarithm

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

$$6^{5x-7}=36$$

Take the logarithm of both sides of the equation.

$$\log(6^{5x-7})=\log(36)$$

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

$$\left(5x-7\right)\log(6)=\log(36)$$

Divide both sides by \log(6).

$$5x-7=\frac{\log(36)}{\log(6)}$$

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

$$5x-7=\log_{6}\left(36\right)$$

Add 7 to both sides of the equation.

$$5x=2-\left(-7\right)$$

Divide both sides by 5.

$$x=\frac{9}{5}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{5\ln(6)}+\frac{9}{5}$$

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