Solve 34^2x-3=15

$$3 { 4 }^{ 2x-3 } = 15$$

Solve for x

$$x = \frac{\log_{2} {(5)} + 6}{4} \approx 2.080482024$$

Steps Using Definition of Logarithm

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

$$3\times 4^{2x-3}=15$$

Divide both sides by 3.

$$4^{2x-3}=5$$

Take the logarithm of both sides of the equation.

$$\log(4^{2x-3})=\log(5)$$

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

$$\left(2x-3\right)\log(4)=\log(5)$$

Divide both sides by \log(4).

$$2x-3=\frac{\log(5)}{\log(4)}$$

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

$$2x-3=\log_{4}\left(5\right)$$

Add 3 to both sides of the equation.

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

Divide both sides by 2.

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

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2\ln(2)}+\frac{\log_{2}\left(5\right)}{4}+\frac{3}{2}$$

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