Solve 10=100^8-5x

$$10 = 100 ^ { 8 - 5 x }$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$100^{8-5x}=10$$

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

$$100^{-5x+8}=10$$

Take the logarithm of both sides of the equation.

$$\log(100^{-5x+8})=\log(10)$$

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

$$\left(-5x+8\right)\log(100)=\log(10)$$

Divide both sides by \log(100).

$$-5x+8=\frac{\log(10)}{\log(100)}$$

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

$$-5x+8=\log_{100}\left(10\right)$$

Subtract 8 from both sides of the equation.

$$-5x=\frac{1}{2}-8$$

Divide both sides by -5.

$$x=-\frac{\frac{15}{2}}{-5}$$

Solve for x (complex solution)

$$x=-\frac{\pi n_{1}i\log(e)}{5}+\frac{3}{2}$$

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