Solve 2.2.25^x+1=15625

$$2.2 .25 ^ { x + 1 } = 15625$$

Solve for x

$$x=\frac{\log_{2}\left(0.0001408\right)}{2}-1\approx -7.397032523$$

Steps Using Definition of Logarithm

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

$$2.2\times 0.25^{x+1}=15625$$

Divide both sides of the equation by 2.2, which is the same as multiplying both sides by the reciprocal of the fraction.

$$0.25^{x+1}=\frac{78125}{11}$$

Take the logarithm of both sides of the equation.

$$\log(0.25^{x+1})=\log(\frac{78125}{11})$$

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

$$\left(x+1\right)\log(0.25)=\log(\frac{78125}{11})$$

Divide both sides by \log(0.25).

$$x+1=\frac{\log(\frac{78125}{11})}{\log(0.25)}$$

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

$$x+1=\log_{0.25}\left(\frac{78125}{11}\right)$$

Subtract 1 from both sides of the equation.

$$x=-\frac{\ln(\frac{78125}{11})}{2\ln(2)}-1$$

Solve for x (complex solution)

$$x=-\frac{i\pi n_{1}}{\ln(2)}+\frac{\log_{2}\left(0.0001408\right)}{2}-1$$

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