# How do you solve for T in 2 = (5.1)^T ?

Sep 4, 2016

$T = 0.4254$

#### Explanation:

As $2 = {5.1}^{T}$

$T = {\log}_{5.1} 2$

= $\log \frac{2}{\log} 5.1$

= $\frac{0.3010}{0.7076}$

= $0.4254$

Sep 4, 2016

$T = \frac{\log \left(2\right)}{\log \left(5.1\right)}$

#### Explanation:

We have: $2 = {\left(5.1\right)}^{T}$

Let's apply logarithms to both sides of the equation:

$\implies \log \left(2\right) = \log \left({5.1}^{T}\right)$

Using the laws of logarithms:

$\implies T \log \left(5.1\right) = \log \left(2\right)$

We can finally solve for $T$ by dividing both sides by $\log \left(5.1\right)$:

$\implies T = \frac{\log \left(2\right)}{\log \left(5.1\right)}$