Solve 4616=4000(1+.098/6)^6t

$$4616 = 4000 ( 1 + \frac { .098 } { 6 } ) ^ { 6 t }$$

Solve for t

$$t = \frac{\log_{\frac{3049}{3000}} {(1.154)}}{6} \approx 1.473477094$$

Steps Using Definition of Logarithm

Divide both sides by 4000.

$$\frac{4616}{4000}=\left(1+\frac{0.098}{6}\right)^{6t}$$

Reduce the fraction \frac{4616}{4000} to lowest terms by extracting and canceling out 8.

$$\frac{577}{500}=\left(1+\frac{0.098}{6}\right)^{6t}$$

Expand \frac{0.098}{6} by multiplying both numerator and the denominator by 1000.

$$\frac{577}{500}=\left(1+\frac{98}{6000}\right)^{6t}$$

Reduce the fraction \frac{98}{6000} to lowest terms by extracting and canceling out 2.

$$\frac{577}{500}=\left(1+\frac{49}{3000}\right)^{6t}$$

Add 1 and \frac{49}{3000} to get \frac{3049}{3000}.

$$\frac{577}{500}=\left(\frac{3049}{3000}\right)^{6t}$$

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

$$\left(\frac{3049}{3000}\right)^{6t}=\frac{577}{500}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{3049}{3000}\right)^{6t})=\log(\frac{577}{500})$$

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

$$6t\log(\frac{3049}{3000})=\log(\frac{577}{500})$$

Divide both sides by \log(\frac{3049}{3000}).

$$6t=\frac{\log(\frac{577}{500})}{\log(\frac{3049}{3000})}$$

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

$$6t=\log_{\frac{3049}{3000}}\left(\frac{577}{500}\right)$$

Divide both sides by 6.

$$t=\frac{\ln(\frac{577}{500})}{6\ln(\frac{3049}{3000})}$$

Solve for t (complex solution)

$$t=\frac{i\pi n_{1}}{3\ln(\frac{3049}{3000})}+\frac{\log_{\frac{3049}{3000}}\left(1.154\right)}{6}$$

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