Solve 5*21^5x=16

$$5 \cdot 21 ^ { 5 x } = 16$$

Solve for x

$$x=\frac{\log_{21}\left(\frac{16}{5}\right)}{5}\approx 0.07640941$$

Steps Using Definition of Logarithm

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

$$5\times 21^{5x}=16$$

Divide both sides by 5.

$$21^{5x}=\frac{16}{5}$$

Take the logarithm of both sides of the equation.

$$\log(21^{5x})=\log(\frac{16}{5})$$

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

$$5x\log(21)=\log(\frac{16}{5})$$

Divide both sides by \log(21).

$$5x=\frac{\log(\frac{16}{5})}{\log(21)}$$

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

$$5x=\log_{21}\left(\frac{16}{5}\right)$$

Divide both sides by 5.

$$x=\frac{\ln(\frac{16}{5})}{5\ln(21)}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{5\ln(21)}-\frac{\log_{21}\left(\frac{5}{16}\right)}{5}$$

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