Solve 15000000=200000{2}^frac{x{48}}

$$15000000 = 200000 { 2 }^{ \frac{ x }{ 48 } }$$

Solve for x

$$x=48\log_{2}\left(75\right)\approx 298.983297144$$

Steps Using Definition of Logarithm

Divide both sides by 200000.

$$\frac{15000000}{200000}=2^{\frac{x}{48}}$$

Divide 15000000 by 200000 to get 75.

$$75=2^{\frac{x}{48}}$$

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

$$2^{\frac{x}{48}}=75$$

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

$$2^{\frac{1}{48}x}=75$$

Take the logarithm of both sides of the equation.

$$\log(2^{\frac{1}{48}x})=\log(75)$$

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

$$\frac{1}{48}x\log(2)=\log(75)$$

Divide both sides by \log(2).

$$\frac{1}{48}x=\frac{\log(75)}{\log(2)}$$

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

$$\frac{1}{48}x=\log_{2}\left(75\right)$$

Multiply both sides by 48.

$$x=\frac{\log_{2}\left(75\right)}{\frac{1}{48}}$$

Solve for x (complex solution)

$$x=\frac{96\pi n_{1}i}{\ln(2)}+48\log_{2}\left(75\right)$$

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