Solve 3700*1.5^x=80000000*2/3

$$3700 \times { 1.5 }^{ x } =80000000 \times \frac{ 2 }{ 3 }$$

Solve for x

$$x=\log_{1.5}\left(\frac{1600000}{111}\right)\approx 23.61728246$$

Steps Using Definition of Logarithm

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

$$3700\times 1.5^{x}=\frac{160000000}{3}$$

Divide both sides by 3700.

$$1.5^{x}=\frac{1600000}{111}$$

Take the logarithm of both sides of the equation.

$$\log(1.5^{x})=\log(\frac{1600000}{111})$$

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

$$x\log(1.5)=\log(\frac{1600000}{111})$$

Divide both sides by \log(1.5).

$$x=\frac{\log(\frac{1600000}{111})}{\log(1.5)}$$

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

$$x=\log_{1.5}\left(\frac{1600000}{111}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.5)}+\log_{1.5}\left(\frac{1600000}{111}\right)$$

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