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Precalculus - Exponential and Logarithmic Functions

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    Study Guide Precalculus – Exponential and Logarithmic Functions 1. Exponential Functions An exponential function has the form: where the base (a) is a positive real number ((a > 0)). Key ideas to remember: • The input (the domain) can be any real number . • The output (the range) is always positive . • Exponential functions grow or decay very quickly compared to linear functions. The Natural Exponential Function One exponential function shows up more often than all the others: the natural exponential function . It is written as: Here, the base (e) is a special number called Euler’s number (pronounced “OIL - er” , not “YOU - ler”). This number is irrational , meaning its decimal: • Never ends • Never repeats Even though the digits may look like they repeat at first, they eventually don’t. Important note: You are not expected to memorize the value of (e), just like you don’t memorize all the digits of ( π ). • Answers can stay written in terms of (e), such as (12e 5 ). • If you are asked to evaluate (e x ), you are allowed to use a calculator.

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    Study Guide Example: Most calculators have an (e x ) key. In some software, this button may be labeled “exp” . Graphs of Exponential Functions To 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 of f ( x ) = 2 x . What the table shows: • When (x) is negative , the output becomes a fraction . • As (x) gets more negative, the graph gets closer and closer to the x - axis , but never touches it. • When (x) is positive , the values grow very quickly. This behavior explains the shape of an exponential growth graph.

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    Study Guide The Anchor Point (0, 1) One very important feature of exponential functions is that: That means every exponential graph passes through the point : This point is called an anchor point , and it’s extremely helpful when sketching or transforming graphs. No matter what the base is: • (2 0 = 1) • (3 0 = 1) • (e 0 = 1) So the point ((0,1)) will always appear. Horizontal Asymptote Exponential graphs also have a horizontal asymptote . For basic exponential functions like (y = a x ), the asymptote is: This means: • The graph gets closer and closer to the x - axis • But it never actually touches or crosses it Transforming Exponential Graphs When 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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