Solve 125=5000(0.01651)(1+0.0165)^t/(1+0.0165)-1

$$125 = \frac { 5000 ( 0.01651 ) ( 1 + 0.0165 ) ^ { t } } { ( 1 + 0.0165 ) - 1 }$$

Solve for t

$$t=\log_{1.0165}\left(\frac{165}{6604}\right)\approx -225.444882628$$

Steps Using Definition of Logarithm

Multiply 5000 and 0.01651 to get 82.55.

$$125=\frac{82.55\left(1+0.0165\right)^{t}}{1+0.0165-1}$$

Add 1 and 0.0165 to get 1.0165.

$$125=\frac{82.55\times 1.0165^{t}}{1+0.0165-1}$$

Add 1 and 0.0165 to get 1.0165.

$$125=\frac{82.55\times 1.0165^{t}}{1.0165-1}$$

Subtract 1 from 1.0165 to get 0.0165.

$$125=\frac{82.55\times 1.0165^{t}}{0.0165}$$

Divide 82.55\times 1.0165^{t} by 0.0165 to get \frac{165100}{33}\times 1.0165^{t}.

$$125=\frac{165100}{33}\times 1.0165^{t}$$

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

$$\frac{165100}{33}\times 1.0165^{t}=125$$

Divide both sides of the equation by \frac{165100}{33}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$1.0165^{t}=\frac{165}{6604}$$

Take the logarithm of both sides of the equation.

$$\log(1.0165^{t})=\log(\frac{165}{6604})$$

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

$$t\log(1.0165)=\log(\frac{165}{6604})$$

Divide both sides by \log(1.0165).

$$t=\frac{\log(\frac{165}{6604})}{\log(1.0165)}$$

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

$$t=\log_{1.0165}\left(\frac{165}{6604}\right)$$