Solve A(t)=500(1+0.7%)^4t=800

$$A ( t ) = 500 ( 1 + 0.7 \% ) ^ { 4 t } = 800$$

Solve for t

$$t = \frac{\log_{1.007} {(1.6)}}{4} \approx 16.844526052$$

Steps Using Definition of Logarithm

Divide both sides by 500.

$$\left(1+\frac{0.7}{100}\right)^{4t}=\frac{800}{500}$$

Reduce the fraction \frac{800}{500} to lowest terms by extracting and canceling out 100.

$$\left(1+\frac{0.7}{100}\right)^{4t}=\frac{8}{5}$$

Expand \frac{0.7}{100} by multiplying both numerator and the denominator by 10.

$$\left(1+\frac{7}{1000}\right)^{4t}=\frac{8}{5}$$

Add 1 and \frac{7}{1000} to get \frac{1007}{1000}.

$$\left(\frac{1007}{1000}\right)^{4t}=\frac{8}{5}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1007}{1000}\right)^{4t})=\log(\frac{8}{5})$$

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

$$4t\log(\frac{1007}{1000})=\log(\frac{8}{5})$$

Divide both sides by \log(\frac{1007}{1000}).

$$4t=\frac{\log(\frac{8}{5})}{\log(\frac{1007}{1000})}$$

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

$$4t=\log_{\frac{1007}{1000}}\left(\frac{8}{5}\right)$$

Divide both sides by 4.

$$t=\frac{\ln(\frac{8}{5})}{4\ln(\frac{1007}{1000})}$$