Solve 7^3x=117.649

$$7 ^ { 3 x } = 117.649$$

Solve for x

$$x=-\log_{7}\left(10\right)+2\approx 0.816705338$$

Steps Using Definition of Logarithm

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

$$7^{3x}=117.649$$

Take the logarithm of both sides of the equation.

$$\log(7^{3x})=\log(117.649)$$

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

$$3x\log(7)=\log(117.649)$$

Divide both sides by \log(7).

$$3x=\frac{\log(117.649)}{\log(7)}$$

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

$$3x=\log_{7}\left(117.649\right)$$

Divide both sides by 3.

$$x=\frac{-3\log_{7}\left(10\right)+6}{3}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{3\ln(7)}-\log_{7}\left(10\right)+2$$

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