Solve 30^x:5^8=6^8

$$30 ^ { x } : 5 ^ { 8 } = 6 ^ { 8 }$$

Solve for x

$$x=8$$

Steps Using Definition of Logarithm

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

$$\frac{1}{390625}\times 30^{x}=1679616$$

Multiply both sides by 390625.

$$30^{x}=656100000000$$

Take the logarithm of both sides of the equation.

$$\log(30^{x})=\log(656100000000)$$

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

$$x\log(30)=\log(656100000000)$$

Divide both sides by \log(30).

$$x=\frac{\log(656100000000)}{\log(30)}$$

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

$$x=\log_{30}\left(656100000000\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(30)}+8$$

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