# How do you solve 5(2^x) = 3 - 2^(x+2)?

Nov 26, 2015

$x = {\log}_{2} \left(\frac{1}{3}\right)$

#### Explanation:

First of all, remember the power rule ${a}^{n} \cdot {a}^{m} = {a}^{n + m}$, we will use it.

Let's transform the equation:

$\textcolor{w h i t e}{\times x} 5 \cdot {2}^{x} = 3 - {2}^{x + 2}$

$\iff 5 \cdot {2}^{x} = 3 - {2}^{x} \cdot {2}^{2}$

$\iff 5 \cdot {2}^{x} = 3 - 4 \cdot {2}^{x}$

... add $4 \cdot {2}^{x}$ on both sides...

$\iff 9 \cdot {2}^{x} = 3$

... divide by $9$ on both sides ...

$\iff {2}^{x} = \frac{1}{3}$

.. apply ${\log}_{2}$ on both sides...

$\iff x = {\log}_{2} \left(\frac{1}{3}\right)$