Solve 1500=200*{2}^frac{x{6.2}}

$$1500 = 200 \times { 2 }^{ \frac{ x }{ 6.2 } }$$

Solve for x

$$x = \frac{31 {(\log_{2} {(15)} - 1)}}{5} \approx 18.022721693$$

Steps Using Definition of Logarithm

Divide both sides by 200.

$$\frac{1500}{200}=2^{\frac{x}{6.2}}$$

Reduce the fraction \frac{1500}{200} to lowest terms by extracting and canceling out 100.

$$\frac{15}{2}=2^{\frac{x}{6.2}}$$

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

$$2^{\frac{x}{6.2}}=\frac{15}{2}$$

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

$$2^{\frac{5}{31}x}=\frac{15}{2}$$

Take the logarithm of both sides of the equation.

$$\log(2^{\frac{5}{31}x})=\log(\frac{15}{2})$$

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

$$\frac{5}{31}x\log(2)=\log(\frac{15}{2})$$

Divide both sides by \log(2).

$$\frac{5}{31}x=\frac{\log(\frac{15}{2})}{\log(2)}$$

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

$$\frac{5}{31}x=\log_{2}\left(\frac{15}{2}\right)$$

Divide both sides of the equation by \frac{5}{31}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\log_{2}\left(15\right)-1}{\frac{5}{31}}$$

Solve for x (complex solution)

$$x=\frac{i\times 62\pi n_{1}}{5\ln(2)}+\frac{31\log_{2}\left(15\right)}{5}-6.2$$

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