# How do you solve 4^x = 1/2?

Apr 1, 2016

$x = - \frac{1}{2}$

#### Explanation:

Take the natural logarithm of both sides of the equation

$\ln | {4}^{x} | = \ln | \frac{1}{2} |$

Rewrite using properties of exponents

$x \ln | 4 | = \ln | \frac{1}{2} |$

$x \ln | {2}^{2} | = \ln | 1 | - \ln | 2 |$

$2 x \ln | 2 | = 0 - \ln | 2 |$

$x = \frac{- \ln | 2 |}{2 \ln | 2 |}$

$x = - \frac{1}{2}$

Apr 3, 2016

#### Explanation:

Rewriting the exponents in a common base:

Don't forget that $\frac{1}{{a}^{n}} = {a}^{-} n$. Thus, $\frac{1}{2} = {2}^{-} 1$.

${\left({2}^{2}\right)}^{x} = {2}^{-} 1$

Using the exponent property ${\left({a}^{n}\right)}^{m} = {a}^{n \times m}$, we get the following:

${2}^{2 x} = {2}^{-} 1$

We can eliminate the bases now and solve the simple linear equation.

$2 x = - 1$

$x = - \frac{1}{2}$

Here are a few helpful exponent rules to know when working with harder problems:

•a^n xx a^m = a^(n + m)

•a^n/a^m = a^(n - m)

•a^(n/m) = root(m)(a^n)

Practice exercises:

1. Solve for x.

${3}^{2 x + 1} \times {9}^{x - 3} = {27}^{4 x - 5}$

$\frac{{4}^{2 x + 5}}{{8}^{3 x - 2}} = {16}^{2 x}$

Challenge Problem:

Solve for x in the equation $\sqrt{12} \times \sqrt[x]{12} = \sqrt[15]{12}$

Good luck!