Solve 5^9=6^x

$$5 ^ { 9 } = 6 ^ { x }$$

Solve for x

$$x=9\log_{6}\left(5\right)\approx 8.084199615$$

Steps Using Definition of Logarithm

Calculate 5 to the power of 9 and get 1953125.

$$1953125=6^{x}$$

Swap sides so that all variable terms are on the left hand side.

$$6^{x}=1953125$$

Take the logarithm of both sides of the equation.

$$\log(6^{x})=\log(1953125)$$

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

$$x\log(6)=\log(1953125)$$

Divide both sides by \log(6).

$$x=\frac{\log(1953125)}{\log(6)}$$

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

$$x=\log_{6}\left(1953125\right)$$

Solve for x (complex solution)

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

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