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Chapter 3: Problem 81

Use a graphing utility and the change-of-base property to graph each function. $$y=\log _{2}(x+2)$$

Short Answer

Expert verified
Using the change-of-base property, the function \(y=\log _{2}(x+2)\) can be rewritten as \(y=\frac{\ln(x+2)}{\ln2}\). After graphing, it appears as a vertical stretch of the function \(y=\ln(x+2)\), with each \(y\)-value divided by \(\ln2\).

Step by step solution

01

Apply the Change-of-Base Formula

Transform the logarithm in base 2 notation (i.e., \(y=\log _{2}(x+2)\)) into a fraction of natural logarithms using the change-of-base formula: \(y=\frac{\ln (x+2)}{\ln 2}\).
02

Plot the Basic Function

Use the graphing utility to sketch the graph of the basic function \(y=\ln (x+2)\).
03

Scale the Graph

Since we are dividing by \(\ln 2\) (which is approximately 0.693), the graph of the original function will be a vertical stretch of the basic function. At each \(x\)-value, the \(y\)-value of the basic function is divided by \(0.693\). Plot these points on the graph.
04

Draw the final graph

Carefully draw the graph that passes through these points and maintains the same overall shape as the graph in step 2.

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