Solve 18^x-2-10=51

$$18 ^ { x - 2 } - 10 = 51$$

Solve for x

$$x=\log_{18}\left(61\right)+2\approx 3.422264749$$

Steps Using Definition of Logarithm

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

$$18^{x-2}-10=51$$

Add 10 to both sides of the equation.

$$18^{x-2}=61$$

Take the logarithm of both sides of the equation.

$$\log(18^{x-2})=\log(61)$$

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

$$\left(x-2\right)\log(18)=\log(61)$$

Divide both sides by \log(18).

$$x-2=\frac{\log(61)}{\log(18)}$$

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

$$x-2=\log_{18}\left(61\right)$$

Add 2 to both sides of the equation.

$$x=\log_{18}\left(61\right)-\left(-2\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(18)}+\log_{18}\left(61\right)+2$$

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