Solve 3*{485}^frac{x{10}}=4000

$$3 \times { 485 }^{ \frac{ x }{ 10 } } = 4000$$

Solve for x

$$x=-10\log_{485}\left(\frac{3}{4000}\right)\approx 11.635291256$$

Steps Using Definition of Logarithm

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

$$3\times 485^{\frac{1}{10}x}=4000$$

Divide both sides by 3.

$$485^{\frac{1}{10}x}=\frac{4000}{3}$$

Take the logarithm of both sides of the equation.

$$\log(485^{\frac{1}{10}x})=\log(\frac{4000}{3})$$

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

$$\frac{1}{10}x\log(485)=\log(\frac{4000}{3})$$

Divide both sides by \log(485).

$$\frac{1}{10}x=\frac{\log(\frac{4000}{3})}{\log(485)}$$

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

$$\frac{1}{10}x=\log_{485}\left(\frac{4000}{3}\right)$$

Multiply both sides by 10.

$$x=\frac{\ln(\frac{4000}{3})}{\frac{1}{10}\ln(485)}$$

Solve for x (complex solution)

$$x=\frac{20\pi n_{1}i}{\ln(485)}-10\log_{485}\left(\frac{3}{4000}\right)$$

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