Solve 5142*1.25^x=100000

$$5142 \times { 1.25 }^{ x } =100000$$

Solve for x

$$x=\log_{1.25}\left(\frac{50000}{2571}\right)\approx 13.299636309$$

Steps Using Definition of Logarithm

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

$$5142\times 1.25^{x}=100000$$

Divide both sides by 5142.

$$1.25^{x}=\frac{50000}{2571}$$

Take the logarithm of both sides of the equation.

$$\log(1.25^{x})=\log(\frac{50000}{2571})$$

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

$$x\log(1.25)=\log(\frac{50000}{2571})$$

Divide both sides by \log(1.25).

$$x=\frac{\log(\frac{50000}{2571})}{\log(1.25)}$$

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

$$x=\log_{1.25}\left(\frac{50000}{2571}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.25)}+\log_{1.25}\left(\frac{50000}{2571}\right)$$

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