Solve 25=510(.5)^x

$$25 = 510 ( .5 ) ^ { x }$$

Solve for x

$$x=-\log_{2}\left(\frac{5}{102}\right)\approx 4.350497247$$

Steps Using Definition of Logarithm

Divide both sides by 510.

$$\frac{25}{510}=0.5^{x}$$

Reduce the fraction \frac{25}{510} to lowest terms by extracting and canceling out 5.

$$\frac{5}{102}=0.5^{x}$$

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

$$0.5^{x}=\frac{5}{102}$$

Take the logarithm of both sides of the equation.

$$\log(0.5^{x})=\log(\frac{5}{102})$$

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

$$x\log(0.5)=\log(\frac{5}{102})$$

Divide both sides by \log(0.5).

$$x=\frac{\log(\frac{5}{102})}{\log(0.5)}$$

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

$$x=\log_{0.5}\left(\frac{5}{102}\right)$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\pi n_{1}}{\ln(2)}-\log_{2}\left(\frac{5}{102}\right)$$

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