Solve 27.5/16=2^frac{x{8}}

$$\frac { 27.5 } { 16 } = 2 ^ { \frac { x } { 8 } }$$

Solve for x

$$x=8\left(\log_{2}\left(55\right)-5\right)\approx 6.250877708$$

Steps Using Definition of Logarithm

Expand \frac{27.5}{16} by multiplying both numerator and the denominator by 10.

$$\frac{275}{160}=2^{\frac{x}{8}}$$

Reduce the fraction \frac{275}{160} to lowest terms by extracting and canceling out 5.

$$\frac{55}{32}=2^{\frac{x}{8}}$$

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

$$2^{\frac{x}{8}}=\frac{55}{32}$$

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

$$2^{\frac{1}{8}x}=1.71875$$

Take the logarithm of both sides of the equation.

$$\log(2^{\frac{1}{8}x})=\log(1.71875)$$

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

$$\frac{1}{8}x\log(2)=\log(1.71875)$$

Divide both sides by \log(2).

$$\frac{1}{8}x=\frac{\log(1.71875)}{\log(2)}$$

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

$$\frac{1}{8}x=\log_{2}\left(1.71875\right)$$

Multiply both sides by 8.

$$x=\frac{\log_{2}\left(55\right)-5}{\frac{1}{8}}$$