Solve 2.5=0.2281.26^x

$$2.5 = 0.228 { 1.26 }^{ x }$$

Solve for x

$$x=\log_{1.26}\left(\frac{625}{57}\right)\approx 10.361657003$$

Steps Using Definition of Logarithm

Divide both sides by 0.228.

$$\frac{2.5}{0.228}=1.26^{x}$$

Expand \frac{2.5}{0.228} by multiplying both numerator and the denominator by 1000.

$$\frac{2500}{228}=1.26^{x}$$

Reduce the fraction \frac{2500}{228} to lowest terms by extracting and canceling out 4.

$$\frac{625}{57}=1.26^{x}$$

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

$$1.26^{x}=\frac{625}{57}$$

Take the logarithm of both sides of the equation.

$$\log(1.26^{x})=\log(\frac{625}{57})$$

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

$$x\log(1.26)=\log(\frac{625}{57})$$

Divide both sides by \log(1.26).

$$x=\frac{\log(\frac{625}{57})}{\log(1.26)}$$

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

$$x=\log_{1.26}\left(\frac{625}{57}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.26)}+\log_{1.26}\left(\frac{625}{57}\right)$$

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