Category Archives: 5.1 Exponential Functions

5.1 Exponential Functions, 6.3 Exponential Functions

Exponents: Simplify

Example:  Rewrite the expression in the form 5^p where p is an algebraic expression.

25/root{3}{5}

Solution:

The original expression
25/root{3}{5}
Rewrite the numerator with a base of 5. 
25=5^2
25/root{3}{5}=5^2/root{3}{5}
 
Rewrite the radical as a fractional exponent.  In general root{n}{x^m}=x^{m/n}.  For this example root{3}{5}= 5^{1/3}
5^2/root{3}{5}=5^2/5^{1/3}
 
Use the quotient rule for exponents to subtract the exponents.
Quotient Rule for Exponents
b^m/b^n=b^{m-n}
5^2/5^{1/3}=5^{2-1/3}
 
Simplify by getting a common denominator and combining the fractions.
5^{2-1/3}=5^{6/3-1/3}=5^{5/3}
 

 

5.1 Exponential Functions, 5.4 Exponential and Logarithmic Equations, 6.6 Logarithmic and Exponential Equations

Solve an Exponential Equation: Relating the Bases

Problem:  Solve the exponential equation.

36^{x}=(1/6)^{x-3}

Solution:

 

The exponential equation
36^{x}=(1/6)^{x-3}
The bases of the exponents on each side of the equation can be made the same.  36 can be written as 6^2 and 1/6 can be written as 6^{-1}
36^{x}=(1/6)^{x-3}
(6^2)^{x}=(6^{-1})^{x-3}

Use the power rule for exponents to multiply the exponents.
Power Rule for Exponents
(b^m)^n=b^{mn}
(6^2)^{x}=(6^{-1})^{x-3}
6^{2x}=6^{-1(x-3)}
6^{2x}=6^{-x+3}
Exponential functions are one-to-one thus giving us the property that if the bases are the same the exponents are equal.
If b^x=b^y then x=y
6^{2x}=6^{-x+3}
2x=-x+3
Solve the remaining equation.  This equation is linear, first get the variables to the same side.
2x=-x+3
2x+x=-x+x+3
3x=3 
Solve:  Get the variable by itself.
3x=3
{3x}/3={3}/3
x=1
Check: Substitute into the original equation.
36^{1}=(1/6)^{1-3}
36=(1/6)^{-2}
36=6^{2}
36=36

The solution to the equation 36^{x}=(1/6)^{x-3} is x=1.

Here is a video example of a similar type of problem.

 

5.1 Exponential Functions, 6.6 Logarithmic and Exponential Equations, Exponential Equations

Solving an Exponential Equation: Relating the Bases

Example:  Solve the exponential equation.

4^{x-9}=1/1024

Solution:

The exponential equation
4^{x-9}=1/1024
Try to write both sides of the equation with the same base.  Try 4 since there is a base of 4 on the left 4^{x-9}=1/4^5
Using a property of negative exponents move the base to the numerator  4^{x-9}=4^{-5}
Now that that the bases are the same the exponents must be equal  4^{x-9}=4^{-5}

x-9=-5
Solve for x
  x-9=-5

x-9+9=-5+9

x=4

The solution the the exponential equation is 4.

Here is a youtube video with a similar example.

 

3.5 Graphing Techiques: Transformations, 5.1 Exponential Functions, 6.3 Exponential Functions, Graphing Exponential Functions

Graphing an Exponential Equation by Transformations

Example: For the function below.  Graph using transformations.  Find the y-intercept.  State the horizontal asymptote and the domain and range.

f(x)=-2^{x+2}+2

First we must examine the base function y=2^x

Graph using plotting points.  We can use the standard set of x-values to find ordered pairs.

xy
-22^(-2)=1/4
-12^(-1)=1/2
02^0=1
12^1=2
22^2=4

The graph below shows the points plotted and the line that connects them.  This graph has a horizontal asymptote at y=0.  The domain is (- infty, infty)  and the range is (0, infty)

 

Analyze the transformations.

f(x)=-2^{x+2}+2

The +2 in the exponent shifts the graph left 2 units.

The – in the front of the base reflects the graph over the x-axis.

The +2 next to the base shifts the graph and the horizontal asymptote up two units.

You can see the graph after the transformations.

The horizontal asymptote is y=2.  The domain is (- infty, infty)  and the range is (- infty, 2)

To find the y-intercept we let x=0.

f(0)=-2^{0+2}+2

f(0)=-(2^{2})+2

f(0)=-(4)+2

f(0)=-4+2

f(0)=-2

Thus the y-intercept is (0,-2)

Here is a youtube video with examples.