What Are Exponential Functions? 

Exponentials are very similar to polynomials and are easily confused. Check out the next two examples.
\(f(x)=x^2\) 
\(g(x)=2^x\)  
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.
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 yaxis. 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. 
Okay, let's watch a video clip that explains this in more detail.
video by MIP4U 

Time for some practice problems.
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 

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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 

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Sketch the graph of \( y = 3^{x2} + 1 \) and find the domain and range.
Problem Statement 

Sketch the graph of \( y = 3^{x2} + 1 \) and find the domain and range.
Solution 

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

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

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Unless otherwise instructed, graph the exponential function \(y=2^x+1\).
Problem Statement 

Unless otherwise instructed, graph the exponential function \(y=2^x+1\).
Solution 

video by PatrickJMT 

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Unless otherwise instructed, graph the exponential function \(y=3^{x}2\).
Problem Statement 

Unless otherwise instructed, graph the exponential function \(y=3^{x}2\).
Solution 

video by PatrickJMT 

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Unless otherwise instructed, graph the exponential function \(y=3(1/2)^x\).
Problem Statement 

Unless otherwise instructed, graph the exponential function \(y=3(1/2)^x\).
Solution 

video by PatrickJMT 

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Unless otherwise instructed, graph the exponential function \(y=3^x1\).
Problem Statement 

Unless otherwise instructed, graph the exponential function \(y=3^x1\).
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 

video by MIP4U 

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Unless otherwise instructed, graph the exponential function \(y=5^x\).
Problem Statement 

Unless otherwise instructed, graph the exponential function \(y=5^x\).
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.
video by MIP4U 

Practice
Solve these exponential functions.
Solve \( 3^{x+2} = 9^{2x3} \)
Problem Statement 

Solve \( 3^{x+2} = 9^{2x3} \)
Solution 

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Solve \( 8^{4x12} = 16^{5x3} \)
Problem Statement 

Solve \( 8^{4x12} = 16^{5x3} \)
Solution 

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Solve \( 27^{3x2} = 81^{2x+7} \)
Problem Statement 

Solve \( 27^{3x2} = 81^{2x+7} \)
Solution 

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Solve \( 3^x = 8 \)
Problem Statement 

Solve \( 3^x = 8 \)
Solution 

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Solve \( e^x = 7 \)
Problem Statement 

Solve \( e^x = 7 \)
Solution 

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Solve \( 5 + 4^{x2} = 23 \)
Problem Statement 

Solve \( 5 + 4^{x2} = 23 \)
Solution 

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

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

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

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

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

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

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

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

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Solve \( 3^{2x}  3^{2x1} = 18 \)
Problem Statement 

Solve \( 3^{2x}  3^{2x1} = 18 \)
Solution 

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Unless otherwise instructed, solve the exponential function \(3^{2x+5}=9^{3x7}\).
Problem Statement 

Unless otherwise instructed, solve the exponential function \(3^{2x+5}=9^{3x7}\).
Solution 

video by MIP4U 

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Unless otherwise instructed, solve the exponential function \( \displaystyle{ \frac{1}{27}=9^{5x7} }\).
Problem Statement 

Unless otherwise instructed, solve the exponential function \( \displaystyle{ \frac{1}{27}=9^{5x7} }\).
Solution 

video by PatrickJMT 

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Unless otherwise instructed, solve the exponential function \( \displaystyle{ \left(\frac{1}{4}\right)^{2x+1}=64 }\).
Problem Statement 

Unless otherwise instructed, solve the exponential function \( \displaystyle{ \left(\frac{1}{4}\right)^{2x+1}=64 }\).
Solution 

video by MIP4U 

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Unless otherwise instructed, solve the exponential function \(3^x + 2\cdot 3^{x+1} = 21\).
Problem Statement 

Unless otherwise instructed, solve the exponential function \(3^x + 2\cdot 3^{x+1} = 21\).
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.
video by PatrickJMT 

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.
Here is a playlist of the videos on this page.
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