Question

Match the logarithmic function with its graph.

$\begin{array}{l}\left(\mathrm{a}\right)f\left(x\right)={\log }_{2}x\left(\mathrm{b}\right)f\left(x\right)={\log }_2\left(-x\right)\\ \left(\mathrm{c}\right)f\left(x\right)=-{\log }_{2}x\left(\mathrm{d}\right)f\left(x\right)=-{\log }_2\left(-x\right)\end{array}$

Short answer

1. The graph of (a) is III.
2. The graph of (b) is II.
3. The graph of (c) is I.
4. The graph of (d) is IV.

Step-by-step solution

Part a. Step 1. Given. 

The given expression is $f\left(x\right)={\log }_{2}x$.

Part a. Step 2. To determine.

We have to find a graph for the given function.

Part a. Step 3. Calculation.

We make a table for $f\left(x\right)={\log }_{2}x$:

x	1	2	4
$f\left(x\right)$	0	1	2

So, the graph is:

This graph matches with graph III.

So, the graph of (a) is III.

Part b. Step 1. Given.

The given expression is $f\left(x\right)={\log }_2\left(-x\right)$.

Part b. Step 2. To determine.

We have to find a graph for the given function.

Part b. Step 3. Calculation.

We make a table for $f\left(x\right)={\log }_2\left(-x\right)$:

x	$-1$	$-2$	$-4$
$f\left(x\right)$	0	1	2

So, the graph is:

This graph matches with graph II.

So, the graph of (b) is II.

Part c. Step 1. Given.

The given expression is $f\left(x\right)=-{\log }_{2}x$.

Part c. Step 2. To determine.

We have to find a graph for the given function.

Part c. Step 3. Calculation.

We make a table for $f\left(x\right)=-{\log }_{2}x$:

x	1	2	4
$f\left(x\right)$	0	$-1$	$-2$

So, the graph is:

This graph matches with graph I.

So, the graph of (c) is I.

Part d. Step 1. Given.

The given expression is $f\left(x\right)=-{\log }_2\left(-x\right)$.

Part d. Step 2. To determine.

We have to find a graph for the given function.

Part d. Step 3. Calculation.

We make a table for :

x	$-1$	$-2$	$-4$
$f\left(x\right)$	0	$-1$	$-2$

So, the graph is:

This graph matches with graph IV.

So, the graph of (d) is IV.