Solve 3.2^3x-2-48=0

$$3.2 ^ { 3 x - 2 } - 48 = 0$$

Solve for x

$$x=-\frac{\log_{3.2}\left(\frac{25}{12288}\right)}{3}\approx 1.776067379$$

Steps Using Definition of Logarithm

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

$$3.2^{3x-2}-48=0$$

Add 48 to both sides of the equation.

$$3.2^{3x-2}=48$$

Take the logarithm of both sides of the equation.

$$\log(3.2^{3x-2})=\log(48)$$

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

$$\left(3x-2\right)\log(3.2)=\log(48)$$

Divide both sides by \log(3.2).

$$3x-2=\frac{\log(48)}{\log(3.2)}$$

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

$$3x-2=\log_{3.2}\left(48\right)$$

Add 2 to both sides of the equation.

$$3x=\frac{\ln(48)}{\ln(\frac{16}{5})}-\left(-2\right)$$

Divide both sides by 3.

$$x=\frac{\frac{\ln(48)}{\ln(\frac{16}{5})}+2}{3}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{3\ln(3.2)}-\frac{\log_{3.2}\left(\frac{25}{12288}\right)}{3}$$

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