Solve 2000=1000^(0.075)(t)

$$2000 = 1000 ^ { ( 0.075 ) ( t ) }$$

Solve for t

$$t = \frac{40 \log_{10} {(2000)}}{9} \approx 14.671244425$$

Steps Using Definition of Logarithm

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

$$1000^{0.075t}=2000$$

Take the logarithm of both sides of the equation.

$$\log(1000^{0.075t})=\log(2000)$$

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

$$0.075t\log(1000)=\log(2000)$$

Divide both sides by \log(1000).

$$0.075t=\frac{\log(2000)}{\log(1000)}$$

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

$$0.075t=\log_{1000}\left(2000\right)$$

Divide both sides of the equation by 0.075, which is the same as multiplying both sides by the reciprocal of the fraction.

$$t=\frac{\log(2000)}{0.075\times 3}$$