Solve 32^3x-2-48=0

$$32 ^ { 3 x - 2 } - 48 = 0$$

Solve for x

$$x = \frac{\log_{2} {(3)} + 14}{15} \approx 1.0389975$$

Steps Using Definition of Logarithm

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

$$32^{3x-2}-48=0$$

Add 48 to both sides of the equation.

$$32^{3x-2}=48$$

Take the logarithm of both sides of the equation.

$$\log(32^{3x-2})=\log(48)$$

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

$$\left(3x-2\right)\log(32)=\log(48)$$

Divide both sides by \log(32).

$$3x-2=\frac{\log(48)}{\log(32)}$$

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

$$3x-2=\log_{32}\left(48\right)$$

Add 2 to both sides of the equation.

$$3x=\frac{\log_{2}\left(48\right)}{5}-\left(-2\right)$$

Divide both sides by 3.

$$x=\frac{\frac{\log_{2}\left(48\right)}{5}+2}{3}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{15\ln(2)}+\frac{\log_{2}\left(3\right)}{15}+\frac{14}{15}$$

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