Solve 0.05=1/2^frac{1{5100}t}

$$0.05 = \frac { 1 } { 2 } ^ { \frac { 1 } { 5100 } t }$$

Solve for t

$$t=5100\left(\log_{2}\left(5\right)+2\right)\approx 22041.833283926$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{2}\right)^{\frac{1}{5100}t}=0.05$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{2}\right)^{\frac{1}{5100}t})=\log(0.05)$$

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

$$\frac{1}{5100}t\log(\frac{1}{2})=\log(0.05)$$

Divide both sides by \log(\frac{1}{2}).

$$\frac{1}{5100}t=\frac{\log(0.05)}{\log(\frac{1}{2})}$$

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

$$\frac{1}{5100}t=\log_{\frac{1}{2}}\left(0.05\right)$$

Multiply both sides by 5100.

$$t=\frac{\log_{2}\left(20\right)}{\frac{1}{5100}}$$