Solve 3*{4.85}^frac{x{10}}=4000

$$3 \times { 4.85 }^{ \frac{ x }{ 10 } } = 4000$$

Solve for x

$$x=-10\log_{4.85}\left(0.00075\right)\approx 45.570198818$$

Steps Using Definition of Logarithm

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

$$3\times 4.85^{\frac{1}{10}x}=4000$$

Divide both sides by 3.

$$4.85^{\frac{1}{10}x}=\frac{4000}{3}$$

Take the logarithm of both sides of the equation.

$$\log(4.85^{\frac{1}{10}x})=\log(\frac{4000}{3})$$

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

$$\frac{1}{10}x\log(4.85)=\log(\frac{4000}{3})$$

Divide both sides by \log(4.85).

$$\frac{1}{10}x=\frac{\log(\frac{4000}{3})}{\log(4.85)}$$

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

$$\frac{1}{10}x=\log_{4.85}\left(\frac{4000}{3}\right)$$

Multiply both sides by 10.

$$x=\frac{\ln(\frac{4000}{3})}{\frac{1}{10}\ln(\frac{97}{20})}$$

Solve for x (complex solution)

$$x=\frac{i\times 20\pi n_{1}}{\ln(4.85)}-10\log_{4.85}\left(0.00075\right)$$

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