# How do you solve for t in 44=2,500times0.5^(t/5.95)?

May 14, 2016

$t \approx 34.68$

#### Explanation:

Given,

$44 = 2500 \cdot {0.5}^{\frac{t}{5.95}}$

Divide both sides by $2500$.

$\frac{44}{2500} = {0.5}^{\frac{t}{5.95}}$

Take the logarithm of both sides since the bases are not the same.

$\log \left(\frac{44}{2500}\right) = \log \left({0.5}^{\frac{t}{5.95}}\right)$

Using the logarithmic property, ${\log}_{\textcolor{p u r p \le}{b}} \left({\textcolor{b l u e}{x}}^{\textcolor{red}{y}}\right) = \textcolor{red}{y} \cdot {\log}_{\textcolor{p u r p \le}{b}} \left(\textcolor{b l u e}{x}\right)$, the equation becomes,

$\log \left(\frac{44}{2500}\right) = \left(\frac{t}{5.95}\right) \log \left(0.5\right)$

$\log \left(\frac{44}{2500}\right) = \log \frac{0.5}{5.95} \cdot t$

Solve for $t$.

$t = \log \frac{\frac{44}{2500}}{\log \frac{0.5}{5.95}}$

$t = \frac{5.95 \log \left(\frac{44}{2500}\right)}{\log \left(0.5\right)}$

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{t \approx 34.68} \textcolor{w h i t e}{\frac{a}{a}} |}}}$