Solve 5120(7/8)^frac{x{2}}=3430

$$5120 ( \frac { 7 } { 8 } ) ^ { \frac { x } { 2 } } = 3430$$

Solve for x

$$x=6$$

Steps Using Definition of Logarithm

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

$$5120\times \left(\frac{7}{8}\right)^{\frac{1}{2}x}=3430$$

Divide both sides by 5120.

$$\left(\frac{7}{8}\right)^{\frac{1}{2}x}=\frac{343}{512}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{7}{8}\right)^{\frac{1}{2}x})=\log(\frac{343}{512})$$

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

$$\frac{1}{2}x\log(\frac{7}{8})=\log(\frac{343}{512})$$

Divide both sides by \log(\frac{7}{8}).

$$\frac{1}{2}x=\frac{\log(\frac{343}{512})}{\log(\frac{7}{8})}$$

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

$$\frac{1}{2}x=\log_{\frac{7}{8}}\left(\frac{343}{512}\right)$$

Multiply both sides by 2.

$$x=\frac{3}{\frac{1}{2}}$$