Solve 25000=10^85^-x

$$25000 = { 10 }^{ 8 } { 5 }^{ -x }$$

Solve for x

$$x=\log_{5}\left(4000\right)\approx 5.15338279$$

Steps Using Definition of Logarithm

Calculate 10 to the power of 8 and get 100000000.

$$25000=100000000\times 5^{-x}$$

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

$$100000000\times 5^{-x}=25000$$

Divide both sides by 100000000.

$$5^{-x}=\frac{25000}{100000000}$$

Reduce the fraction \frac{25000}{100000000} to lowest terms by extracting and canceling out 25000.

$$5^{-x}=\frac{1}{4000}$$

Take the logarithm of both sides of the equation.

$$\log(5^{-x})=\log(\frac{1}{4000})$$

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

$$-x\log(5)=\log(\frac{1}{4000})$$

Divide both sides by \log(5).

$$-x=\frac{\log(\frac{1}{4000})}{\log(5)}$$

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

$$-x=\log_{5}\left(\frac{1}{4000}\right)$$

Divide both sides by -1.

$$x=-\frac{\log_{5}\left(4000\right)}{-1}$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(5)}+5\log_{5}\left(2\right)+3$$

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