Solve for x: log₄x = -2.5

Steps to Solve

1. Convert the mixed number to a decimal

Convert $-2 \frac{1}{2}$ to a decimal.

$-2 \frac{1}{2} = -2.5$

So, the equation becomes:

$\log_4 x = -2.5$

2. Convert the logarithmic equation to exponential form

The logarithmic equation $\log_b a = c$ is equivalent to the exponential equation $b^c = a$.

In our case, $\log_4 x = -2.5$, so $b = 4$, $c = -2.5$ and $a = x$. Therefore, the exponential form is:

$4^{-2.5} = x$

3. Simplify the exponential expression

Rewrite $4$ as $2^2$ and $-2.5$ as $-\frac{5}{2}$

$x = 4^{-2.5} = (2^2)^{-\frac{5}{2}}$

Using the power of a power rule $(a^m)^n = a^{mn}$:

$x = 2^{2 \cdot (-\frac{5}{2})} = 2^{-5}$

4. Calculate the final value of x

$x = 2^{-5} = \frac{1}{2^5} = \frac{1}{32}$