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Precalculus - Exponential and Logarithmic Functions

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Study GuidePrecalculus–Exponential and Logarithmic Functions1.Exponential FunctionsAnexponential functionhas the form:where the base (a) is apositive real number((a > 0)).Key ideas to remember:•Theinput(the domain) can beany real number.•Theoutput(the range) is alwayspositive.•Exponential functions grow or decay very quickly compared to linear functions.The Natural Exponential FunctionOne exponential function shows up more often than all the others: thenatural exponential function.It is written as:Here, the base (e) is a special number calledEuler’s number(pronounced“OIL-er”, not “YOU-ler”).This number isirrational, meaning its decimal:•Never ends•Never repeatsEven though the digits maylooklike they repeat at first, they eventually don’t.Important note:You arenot expected to memorizethe value of (e), just like you don’t memorize all the digits of (π).•Answers can stay written in terms of (e), such as (12e5).•If you are asked to evaluate (ex), you are allowed to use a calculator.

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Study GuideExample:Most calculators have an (ex) key. In some software, this button may be labeled“exp”.Graphs of Exponential FunctionsTo understand exponential graphs, let’s look at a common example:If we plug in some integer values for (x), we get:Figure 1 The graph off(x) = 2x.What the table shows:•When (x) isnegative, the output becomes afraction.•As (x) gets more negative, the graph gets closer and closer to thex-axis, but never touchesit.•When (x) ispositive, the values grow very quickly.This behavior explains the shape of an exponential growth graph.

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Study GuideThe Anchor Point (0, 1)One very important feature of exponential functions is that:That meansevery exponential graph passes through the point:This point is called ananchor point, and it’s extremely helpful when sketching or transforminggraphs.No matter what the base is:•(20= 1)•(30= 1)•(e0= 1)So the point ((0,1)) will always appear.Horizontal AsymptoteExponential graphs also have ahorizontal asymptote.For basic exponential functions like (y = ax), the asymptote is:This means:•The graph gets closer and closer to the x-axis•But it never actually touches or crosses itTransforming Exponential GraphsWhen an exponential function is transformed, we use the anchor point ((0,1)) to help track changes.Example 1:Sketch the graph of
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Document Details

University
Texas A&M University
Course
Precalculus
Subject
Mathematics

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