Solve 100*2^4x=15

$$100 \cdot 2 ^ { 4 x } = 15$$

Solve for x

$$x=\frac{\log_{2}\left(\frac{3}{5}\right)}{4}-\frac{1}{2}\approx -0.684241399$$

Steps Using Definition of Logarithm

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

$$100\times 2^{4x}=15$$

Divide both sides by 100.

$$2^{4x}=\frac{3}{20}$$

Take the logarithm of both sides of the equation.

$$\log(2^{4x})=\log(\frac{3}{20})$$

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

$$4x\log(2)=\log(\frac{3}{20})$$

Divide both sides by \log(2).

$$4x=\frac{\log(\frac{3}{20})}{\log(2)}$$

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

$$4x=\log_{2}\left(\frac{3}{20}\right)$$

Divide both sides by 4.

$$x=\frac{\ln(\frac{3}{20})}{4\ln(2)}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2\ln(2)}+\frac{\log_{2}\left(\frac{3}{20}\right)}{4}$$

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