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17Calculus Precalculus - Exponential Functions

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Single Variable Calculus
Derivatives
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Precalculus
Functions

Exponential Functions

What Are Exponential Functions?

Exponentials are very similar to polynomials and are easily confused. Check out the next two examples.

\(f(x)=x^2\)

\(g(x)=2^x\)

polynomial

exponential

The polynomial you should be familiar with. Notice that the difference between a polynomial and an exponential is where the variable is, in the base or in the exponent. In the base, we have a polynomial, in the exponent, we have an exponential function.

In general, an exponential function is written \(f(x) = Ca^x\) where a is a positive real number (\(a \gt 0\)) and \(a \neq 1 \) and C is a real number.
a is called the growth factor
C is called the initial value since \(f(0) = Ca^0 = C\)

Remember these laws of exponents? These also hold for exponential functions.

1.

\(a^s \cdot a^t = a^{s+t}\)

2.

\((a^s)^t = a^{st}\)

3.

\( (ab)^s = a^s \cdot b^s \)

4.

\( 1^s = 1 \)

5.

\(\displaystyle{ a^{-s} = \frac{1}{a^s} = \left[ \frac{1}{a} \right]^s }\)

6.

\( a^0 = 1 \) since \(a\neq0\)

7.

if \(a^u = a^v\) then \(u=v\)

Also remember that \((a+b)^s \neq a^s + b^s\). An example of this is \((a+b)^2 = a^2 +2ab + b^2 \neq a^2 + b^2\).
Before we go on, here is a good video that shows several examples so that you can get a feel for exponential functions.

PatrickJMT - Identify the Exponential Function [2min-34secs]

video by PatrickJMT

Notice we are calling \(g(x)=a^x\) a function. The use of the word 'function' is deliberate since exponentials pass the vertical line test. Let's look at a few exponential functions. We will draw them on the same graph so that we can compare them.

In plot 1 below, notice that all of the exponential functions have the same general shape and that they all go through the point \((0,1)\). This makes sense, since \(2^0=3^0=4^0=1\). This shape holds for exponential functions \(y=a^x\) where \(a>1\).

Now take a careful look at plot 2. Notice that the exponential functions are still in the form \(y=a^x\) but in these cases \(0 < a < 1\). Said another way, \(a > 1\) and \(y=(1/a)^x\). And said still another way, \(a > 1\) and \(y=a^{-x}\).

Again, all plots go through the point \((0,1)\) and they all have the same general shape.

When comparing the two plots, notice that they are flipped with respect to the y-axis. These are the kind of things you need to look at and think about in calculus when you are given a graph.

plot 1: \(\color{red}{y=2^x}\); \(\color{blue}{y=3^x}\); \(\color{green}{y=4^x}\)

plot 2: \(\color{red}{y=(1/2)^x}\); \(\color{blue}{y=(1/3)^x}\); \(\color{green}{y=(1/4)^x}\)

Study Tip

When given alternate ways of describing an equation like we have done here, focus on the one that makes most sense to you and then determine the relationship between the one you chose and the other ones. Learning more than one at time can be overwhelming and an unnecessary waste of your time and mental energy.
For this set of equations, I think the easiest are \(y=a^x\) and \(y=a^{-x}\), for \(a > 1\) in both cases.

Okay, let's watch a video clip that explains this in more detail.

MIP4U - Graph Exponential Functions [7min-55secs]

video by MIP4U

Okay, time for the practice problems.

You Can Have an Amazing Memory: Learn Life-Changing Techniques and Tips from the Memory Maestro

Practice

Unless otherwise instructed, graph these exponential functions.

Sketch the graph of \( y=2^x \) and find the domain and range.

Problem Statement

Sketch the graph of \( y=2^x \) and find the domain and range.

Solution

2987 video solution

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Sketch the graph of \(y=(1/3)^x \) and find the domain and range.

Problem Statement

Sketch the graph of \(y=(1/3)^x \) and find the domain and range.

Solution

2988 video solution

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Sketch the graph of \( y = 3^{x-2} + 1 \) and find the domain and range.

Problem Statement

Sketch the graph of \( y = 3^{x-2} + 1 \) and find the domain and range.

Solution

2989 video solution

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Sketch the graph of \( y= 5 - 2^{3-x} \) and find the domain and range.

Problem Statement

Sketch the graph of \( y= 5 - 2^{3-x} \) and find the domain and range.

Solution

2990 video solution

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\(y=2^x+1\).

Problem Statement

Graph \(y=2^x+1\).

Solution

PatrickJMT - 1693 video solution

video by PatrickJMT

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\(y=3^{-x}-2\)

Problem Statement

Solve \(y=3^{-x}-2\)

Solution

PatrickJMT - 1694 video solution

video by PatrickJMT

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\(y=3(1/2)^x\).

Problem Statement

Graph \(y=3(1/2)^x\).

Solution

PatrickJMT - 1695 video solution

video by PatrickJMT

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\(y=-3^x-1\).

Problem Statement

Graph \(y=-3^x-1\).

Solution

PatrickJMT - 1696 video solution

video by PatrickJMT

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Determine the equation of this transformed exponential function.

Problem Statement

Determine the equation of this transformed exponential function.

Solution

MIP4U - 1697 video solution

video by MIP4U

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\(y=5^x\).

Problem Statement

Graph \(y=5^x\).

Solution

Khan Academy - 1690 video solution

video by Khan Academy

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Special Exponential Function

There is a special exponential function that is used extensively in calculus, \(f(x)=e^x\). The number \(e \approx 2.7183\) is a special irrational number. (You need calculus to understand where it comes from.) Since \(e > 1\), the graph looks like plot 1 above.

Solving Exponential Equations

In order to get a feel for working with exponential functions, you may be asked to solve exponential functions. The first technique does not require the knowledge of logarithms. This next video contains an explanation how to do this followed by lots of examples.

MIP4U - Solving Exponential Equations [6min-27secs]

video by MIP4U

Practice

Solve these exponential functions.

\( 3^{x+2} = 9^{2x-3} \)

Problem Statement

Solve \( 3^{x+2} = 9^{2x-3} \)

Solution

2991 video solution

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\( 8^{4x-12} = 16^{5x-3} \)

Problem Statement

Solve \( 8^{4x-12} = 16^{5x-3} \)

Solution

2992 video solution

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\( 27^{3x-2} = 81^{2x+7} \)

Problem Statement

Solve \( 27^{3x-2} = 81^{2x+7} \)

Solution

2993 video solution

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\( 3^x = 8 \)

Problem Statement

Solve \( 3^x = 8 \)

Solution

2994 video solution

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\( e^x = 7 \)

Problem Statement

Solve \( e^x = 7 \)

Solution

2995 video solution

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\( 5 + 4^{x-2} = 23 \)

Problem Statement

Solve \( 5 + 4^{x-2} = 23 \)

Solution

2996 video solution

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\( 3 + 2e^{3-x} = 7 \)

Problem Statement

Solve \( 3 + 2e^{3-x} = 7 \)

Solution

2997 video solution

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\( 3^{x^2+4x} = 1/27 \)

Problem Statement

Solve \( 3^{x^2+4x} = 1/27 \)

Solution

2998 video solution

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\( 2^{x^2} \cdot 2^{3x} = 16 \)

Problem Statement

Solve \( 2^{x^2} \cdot 2^{3x} = 16 \)

Solution

2999 video solution

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\( 4^{2x} - 20 \cdot 4^x+ 64 = 0 \)

Problem Statement

Solve \( 4^{2x} - 20 \cdot 4^x+ 64 = 0 \)

Solution

3000 video solution

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\( 3^{2x} - 3^{2x-1} = 18 \)

Problem Statement

Solve \( 3^{2x} - 3^{2x-1} = 18 \)

Solution

3001 video solution

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\(3^{2x+5}=9^{3x-7}\).

Problem Statement

Solve \(3^{2x+5}=9^{3x-7}\).

Solution

MIP4U - 1691 video solution

video by MIP4U

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\( \displaystyle{ \frac{1}{27}=9^{5x-7} }\).

Problem Statement

Solve \( \displaystyle{ \frac{1}{27}=9^{5x-7} }\).

Solution

PatrickJMT - 1689 video solution

video by PatrickJMT

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\( \displaystyle{ \left(\frac{1}{4}\right)^{2x+1}=64 }\)

Problem Statement

Solve \( \displaystyle{ \left(\frac{1}{4}\right)^{2x+1}=64 }\)

Solution

MIP4U - 1692 video solution

video by MIP4U

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\(3^x + 2\cdot 3^{x+1} = 21\)

Problem Statement

Solve \(3^x + 2\cdot 3^{x+1} = 21\)

Solution

Dr Chris Tisdell - 2127 video solution

video by Dr Chris Tisdell

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Rules of Exponents - Review

Here are some videos to help you review rules of exponents. These videos also contain plenty of examples.

PatrickJMT - Basic Exponent Properties [12min-26secs]

video by PatrickJMT

MIP4U - Simplify Exponential Expressions With Negative Exponents - Basic [4min-44secs]

video by MIP4U

Next

Exponentials are very closely related to logarithms. So, after working the practice problems on this page, your next logical step is logarithms. After that, we will discuss how to use exponentials and logarithms together and then solve some application problems.

Really UNDERSTAND Precalculus

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