Q 58. - Precalculus Enhanced with Graphing Utilities

Question

Find  $\text{'}a\text{'}$so that the graph of $f\left(x\right)={\log }_{a}x$  contains the point  $(\frac{1}{2},-4)$

Step-by-Step Solution

Answer

The value of $\text{'}a\text{'}$ is $-4$

Step 1. Given information

Given function is  $f\left(x\right)={\log }_a\left(x\right)$

Step 2. Omitting logaritm

Omitting logarithm, we have, 

$x=a^y$

Now, the function holds the point $(\frac{1}{2},-4)$

Hence, putting the point to the function we have, 

$\frac{1}{2}=a^{-4}$

Simplifying, we have,

$a=2^{\frac{1}{4}}$

Step 3. Check

Now, for $a=2^{\frac{1}{4}}$ 

we have,

$$\begin{gathered} {\log }_{2^{\frac{1}{4}}}\left(\frac{1}{2}\right) \\ =-4 \end{gathered}$$