Solve 9(4^3x-2)-7=3

$$9 ( 4 ^ { 3 x - 2 } ) - 7 = 3$$

Solve for x

$$x=\frac{\log_{2}\left(\frac{5}{9}\right)+5}{6}\approx 0.692000516$$

Steps Using Definition of Logarithm

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

$$9\times 4^{3x-2}-7=3$$

Add 7 to both sides of the equation.

$$9\times 4^{3x-2}=10$$

Divide both sides by 9.

$$4^{3x-2}=\frac{10}{9}$$

Take the logarithm of both sides of the equation.

$$\log(4^{3x-2})=\log(\frac{10}{9})$$

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

$$\left(3x-2\right)\log(4)=\log(\frac{10}{9})$$

Divide both sides by \log(4).

$$3x-2=\frac{\log(\frac{10}{9})}{\log(4)}$$

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

$$3x-2=\log_{4}\left(\frac{10}{9}\right)$$

Add 2 to both sides of the equation.

$$3x=\frac{\ln(\frac{10}{9})}{2\ln(2)}-\left(-2\right)$$

Divide both sides by 3.

$$x=\frac{\frac{\ln(\frac{10}{9})}{2\ln(2)}+2}{3}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{3\ln(2)}+\frac{\log_{2}\left(\frac{5}{9}\right)}{6}+\frac{5}{6}$$

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