Solve 47=610(0.5)^frac{x{4}}

$$47 = 610 ( 0.5 ) ^ { \frac { x } { 4 } }$$

Solve for x

$$x=4\log_{2}\left(\frac{610}{47}\right)\approx 14.792306323$$

Steps Using Definition of Logarithm

Divide both sides by 610.

$$\frac{47}{610}=0.5^{\frac{x}{4}}$$

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

$$0.5^{\frac{x}{4}}=\frac{47}{610}$$

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

$$0.5^{\frac{1}{4}x}=\frac{47}{610}$$

Take the logarithm of both sides of the equation.

$$\log(0.5^{\frac{1}{4}x})=\log(\frac{47}{610})$$

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

$$\frac{1}{4}x\log(0.5)=\log(\frac{47}{610})$$

Divide both sides by \log(0.5).

$$\frac{1}{4}x=\frac{\log(\frac{47}{610})}{\log(0.5)}$$

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

$$\frac{1}{4}x=\log_{0.5}\left(\frac{47}{610}\right)$$

Multiply both sides by 4.

$$x=-\frac{\frac{\ln(\frac{47}{610})}{\ln(2)}}{\frac{1}{4}}$$

Solve for x (complex solution)

$$x=-\frac{i\times 8\pi n_{1}}{\ln(2)}+4\log_{2}\left(\frac{610}{47}\right)$$

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