Solve 3*2^5x=1

$$3 \cdot 2 ^ { 5 x } = 1$$

Solve for x

$$x=-\frac{\log_{2}\left(3\right)}{5}\approx -0.3169925$$

Steps Using Definition of Logarithm

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

$$3\times 2^{5x}=1$$

Divide both sides by 3.

$$2^{5x}=\frac{1}{3}$$

Take the logarithm of both sides of the equation.

$$\log(2^{5x})=\log(\frac{1}{3})$$

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

$$5x\log(2)=\log(\frac{1}{3})$$

Divide both sides by \log(2).

$$5x=\frac{\log(\frac{1}{3})}{\log(2)}$$

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

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

Divide both sides by 5.

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

Solve for x (complex solution)

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

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