Question

Let g(x)=log_2x1. find g(5)2. find g(-3)3. find g^-1(x)4. find g^-1(-3)

Answer 1

$g(x)=\log _2x$

You evaluate the equation in the given values of x:

$\log _ba=\frac{\log a}{\log b}$

1. g(5)$\begin{gathered} g(5)=\log _25 \\ g(5)=\frac{\log 5}{\log 2}=2.321 \end{gathered}$$g(5)=2.321$2. g(-3)$\begin{gathered} g(-3)=\log _2(-3) \\ g(-3)=\frac{\log (-3)}{\log 2}=\text{undefined} \end{gathered}$

The logarithm of a negative number is undefined

3. g^-1(x)

To find the inverse function you:

-write the function with x and y:

$\begin{gathered} g(x)=\log _2x \\ y=\log _2x \end{gathered}$

-Solve variable x:

knowing that:

$\begin{gathered} \log _ba=c \\ b^c=a \end{gathered}$$\begin{gathered} y=\log _2x \\ \\ 2^y=x \end{gathered}$

- Change the x for (g^-1(x)) and the y for x:

$g^{-1}(x)=2^x$4.g^-1(-3)​

As:

$n^{-m}=\frac{1}{n^m}$

$\begin{gathered} g^{-1}(-3)=2^{-3} \\ \\ =\frac{1}{2^3}=\frac{1}{8} \end{gathered}$