Solve 2/5^3x=100000=

$$2 / 5 ^ { 3 x } = 100000 =$$

Solve for x

$$x=\frac{5\log_{\frac{2}{5}}\left(10\right)}{3}\approx -4.188235991$$

Steps Using Definition of Logarithm

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

$$\left(\frac{2}{5}\right)^{3x}=100000$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{2}{5}\right)^{3x})=\log(100000)$$

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

$$3x\log(\frac{2}{5})=\log(100000)$$

Divide both sides by \log(\frac{2}{5}).

$$3x=\frac{\log(100000)}{\log(\frac{2}{5})}$$

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

$$3x=\log_{\frac{2}{5}}\left(100000\right)$$

Divide both sides by 3.

$$x=\frac{5\ln(10)}{3\ln(\frac{2}{5})}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3\ln(\frac{2}{5})}+\frac{5\log_{\frac{2}{5}}\left(10\right)}{3}$$

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