Solve 7000^0.15x=21000

$${ 7000 }^{ 0.15x } = 21000$$

Solve for x

$$x = \frac{20 {(\log_{7000} {(3)} + 1)}}{3} \approx 7.493903926$$

Steps Using Definition of Logarithm

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

$$7000^{0.15x}=21000$$

Take the logarithm of both sides of the equation.

$$\log(7000^{0.15x})=\log(21000)$$

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

$$0.15x\log(7000)=\log(21000)$$

Divide both sides by \log(7000).

$$0.15x=\frac{\log(21000)}{\log(7000)}$$

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

$$0.15x=\log_{7000}\left(21000\right)$$

Divide both sides of the equation by 0.15, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\log_{7000}\left(21000\right)}{0.15}$$

Solve for x (complex solution)

$$x=\frac{i\times 40\pi n_{1}}{3\ln(7000)}+\frac{20\log_{7000}\left(3\right)}{3}+\frac{20}{3}$$

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