Solve 138*2^2x=46

$$138 \cdot 2 ^ { 2 x } = 46$$

Solve for x

$$x=-\frac{\log_{2}\left(3\right)}{2}\approx -0.79248125$$

Steps Using Definition of Logarithm

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

$$138\times 2^{2x}=46$$

Divide both sides by 138.

$$2^{2x}=\frac{1}{3}$$

Take the logarithm of both sides of the equation.

$$\log(2^{2x})=\log(\frac{1}{3})$$

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

$$2x\log(2)=\log(\frac{1}{3})$$

Divide both sides by \log(2).

$$2x=\frac{\log(\frac{1}{3})}{\log(2)}$$

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

$$2x=\log_{2}\left(\frac{1}{3}\right)$$

Divide both sides by 2.

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

Solve for x (complex solution)

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

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