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2A. Exponential and Logarithmic Functions
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AP Precalculus

2.4 Exponential function manipulation

Lesson
Worksheet
Lesson

Introduction

Learning objective

  • 2.4.A Rewrite exponential expressions in equivalent forms.

Product porperty of exponents

We have previously learned about the product property for exponents. Let's review where it comes from.

Consider the expression a^{5} \cdot a^{3}. Let's think abou what it would look like if we expanded it:

The image shows 8 a's being multiplied by each other.

We can see how this could be rewritten as a^8. Notice, 8 is also the sum of the powers in the original expression.

So, in our example above, \begin{aligned}a^{5}\cdot a^{3} &= a^{5+3}\\&=a^{8}\end{aligned}

This leads us to the product property for exponents:

b^{m} \cdot b^{n}=b^{m+n}

That is, when multiplying terms with a common base:

  • Keep the same base

  • Find the sum of the exponents

In other words, when multiplying terms with like bases, we add the exponents.

We can visualize this graphically as a transformation. Consider a simple exponential function: f\left(x\right)=2^x.

If we apply a translation to the left 3 units we can write the function as f\left(x\right)=2^{\left(x+3\right)}.

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f\left(x\right)=2^x
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f\left(x+3\right)=2^{\left(x+3\right)}

Applying the product property we can rewrite the function as f\left(x+3\right)=2^{\left(x+3\right)}=2^x\cdot 2^3=8\cdot2^x

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Looking at this on the graph we can see that it has the same graph as f\left(x+3\right)=2^{\left(x+3\right)}.

We can conclude that for this function f\left(x\right) a translation to the left 3 units is equivalent to a vertical dilation by a factor of 8.

More generally we can say that for an exponential function f\left(x\right)=b^x a horizontal translation f\left(x+k\right)=b^{\left(x+k\right)} is equivalent to a vertical dilation f\left(x\right)=b^{\left(x+k\right)}=b^x\cdot b^k=ab^x, where a=b^k.

Examples

Example 1

Fill in the blank to make the eqution true: b^{2}\cdot b^{⬚} = b^{2 + 3}

Worked Solution
Create a strategy

We can use the exponent law: a^{m} \cdot a^{n}=a^{m+n}

Apply the idea

Since we know that we can add powers when two common bases are being multiplied together, we know that the blank box must be the power that is being added to 2 on the right hand side of the equation. Therefore, we know that 3 must go into the blank box to make the equation true. b^{2}\cdot b^{3} = b^{2 + 3}

Example 2

Simplify m^{2} \cdot m^{7} + r^{3} \cdot r^{2}, giving your answer in exponential form.

Worked Solution
Create a strategy

We can use the exponent law: a^{m} \cdot a^{n}=a^{m+n}

Apply the idea
\displaystyle m^{2} \cdot m^{7} + r^{3} \cdot r^{2}\displaystyle =\displaystyle m^{2+7} + r^{3+2}Add the powers of the bases m and r
\displaystyle =\displaystyle m^{9} + r^{5}Simplify the powers

Example 3

Consider the function: h\left(x\right)=2^{\left(x-4\right)}

a

Identify the transformation on the parent function that produced h\left(x\right).

Worked Solution
Create a strategy

Consider how h\left(x\right)=2^{\left(x-4\right)} is different from the parent function f\left(x\right)=2^x.

Apply the idea

A translation to the right 4 units.

b

Rewrite h\left(x\right) to use a vertical dilation instead of a translation.

Worked Solution
Create a strategy

Use the product property of exponents to rewrite h\left(x\right).

Apply the idea

We will use the fact that b^{m} \cdot b^{n}=b^{m+n}

\displaystyle h\left(x\right)\displaystyle =\displaystyle 2^{\left(x-4\right)}
\displaystyle =\displaystyle 2^{\left(x+\left(-4\right)\right)}Rewrite subtraction as addition
\displaystyle =\displaystyle 2^x\cdot 2^{-4}Product property of exponents
\displaystyle =\displaystyle 2^x\cdot \dfrac{1}{2^4}Negative exponent property
\displaystyle =\displaystyle \dfrac{1}{16}\cdot2^xEvaluate the exponent
Reflect and check

This shows that a translation of h\left(x\right) to the right 4 units is equivalent to a vertical dilation of h\left(x\right) by a factor of \dfrac{1}{16}.

Idea summary

We can use the product property of exponents: b^{m} \cdot b^{n}=b^{m+n} to rewrite a horizontal translation as a vertical dilation and vice versa.

Power property of exponents

We have previously learned about the power property for exponents. Let's review where it comes from.

Consider the expression (a^{2})^{3}. What is the resulting power of base a? To find out, have a look at the expanded form of the expression:

\displaystyle (a^{2})^{3}\displaystyle =\displaystyle (a^{2})\times (a^{2}) \times (a^{2})
\displaystyle =\displaystyle (a \times a) \times (a \times a) \times (a \times a)
\displaystyle =\displaystyle a \times a \times a \times a \times a \times a
\displaystyle =\displaystyle a^{6}

In the expanded form, we can see that we are multiplying six groups of a together. That is, \\(a^{2})^{3}=a^{6}.

We can confirm this result using the product property for exponents:

\displaystyle (a^{2}) \times (a^{2}) \times (a^{2})\displaystyle =\displaystyle a^{2+2+2}
\displaystyle =\displaystyle a^6

This leads us to the power property for exponents:

(b^{m})^{n} = b^{m\cdot n}

That is, when simplifying a term with a power that itself has a power:

  • Keep the same base

  • Multiply the exponents

We can visualize this graphically as a transformation. Consider again the simple exponential function: f\left(x\right)=2^x.

If we apply a horizontal dilation by a factor of 3 we can write the function as f\left(3x\right)=2^{\left(3x\right)}.

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f\left(x\right)=2^x
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f\left(3x\right)=2^{\left(3x\right)}

Applying the power property we can rewrite the function as f\left(3x\right)=2^{\left(3x\right)}={\left(2^3\right)}^x=8^x

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Looking at this on the graph we can see that it has the same graph as f\left(3x\right)=2^{\left(3x\right)}.

We can conclude that for this function f\left(x\right) a horizontal dilation by a factor of 3 is equivalent to changing the base to 8.

More generally we can say that for an exponential function f\left(x\right)=b^x a horizonal dilation f\left(kx\right)=b^{\left(kx\right)} is equivalent to a change of base f\left(x\right)={b^c}^x, where b^c is a constant.

Examples

Example 4

Simplify \left(a^{5}\right)^{3}.

Worked Solution
Create a strategy

Use the power of a power law.

Apply the idea
\displaystyle \left(a^{5}\right)^{3}\displaystyle =\displaystyle a^{5 \cdot 3}Multiply the powers of the variable
\displaystyle =\displaystyle a^{15}Evaluate the multiplication

Example 5

Simplify (a^{9}\cdot b^{3})^{4}

Worked Solution
Create a strategy

We can use the rule: (ab)^{n}=a^{n}b^{n}

Apply the idea
\displaystyle (a^{9}\cdot b^{3})^{4}\displaystyle =\displaystyle (a^{9})^{4}(b^{3})^{4}Start with the power of a product rule
\displaystyle =\displaystyle a^{9\cdot 4}\cdot b^{3\cdot 4}Multiply the powers
\displaystyle =\displaystyle a^{36}b^{12}Evaluate the powers

Example 6

Simplify \left(-2x^{2}\right)^{2}.

Worked Solution
Create a strategy

Use the power of a power law.

Apply the idea
\displaystyle \left(-2x^{2}\right)^{2}\displaystyle =\displaystyle \left(-2\right)^{2}x^{2\cdot2}Multiply the powers of the variable
\displaystyle =\displaystyle 4x^4Evaluate the multiplication and coefficient
Idea summary

We can use the power property of exponents: \left(b^m\right)^m=b^{m\cdot n} to rewrite a horizontal dilation as an exponential function with a different base.

Negative exponents and unit fractions

The negative exponent property states: b^{-n}=\dfrac{1}{b^{n}}

This can be understood as a 'flipping' operation, where the base moves from the numerator to the denominator, or vice versa, when the sign of the exponent changes. In other words, if you have a negative exponent, you can make it positive by taking the reciprocal of the base.

Similarly, an exponential expression can involve what is called an exponential unit fraction. This is when the exponent is the reciprocal of a natural number (positive whole number), denoted as \dfrac{1}{k}. An exponential unit fraction takes the form b^{\frac{1}{k}}.

Let's consider the example x^{\frac{1}{2}}:

\displaystyle x^{\frac{2}{2}}\displaystyle =\displaystyle x
\displaystyle \left(x^{\frac{1}{2}}\right)^{2}\displaystyle =\displaystyle x
\displaystyle \left(x^{\frac{1}{2}}\right)^{2}\displaystyle =\displaystyle \left(\sqrt{x}\right)^{2}
\displaystyle x^{\frac{1}{2}}\displaystyle =\displaystyle \sqrt{x}

An exponential unit fraction, b^{\frac{1}{k}}, represents the kth root of the base b.

Examples

Example 7

Express 3x^{-3} with a positive exponent.

Worked Solution
Create a strategy

We can use the negative exponent property: b^{-n}=\dfrac{1}{b^{n}}.

Apply the idea
\displaystyle 3x^{-3}\displaystyle =\displaystyle 3 \cdot x^{-3}Separate the coefficient and the term with the power
\displaystyle =\displaystyle 3 \cdot \dfrac{1}{x^{3}}Negative exponent property
\displaystyle =\displaystyle \dfrac{3}{x^{3}}Multiply
Reflect and check

Since the power is only on the x, the value of the coefficient hasn't changed.

Example 8

Consider the following.

a

Rewrite x^{\frac{1}{3}} in radical form.

Worked Solution
Create a strategy

Use the rule b^{\frac{1}{k}}=\sqrt[k]{b}.

Apply the idea

x^{\frac{1}{3}}=\sqrt[3]{x}

b

Evaluate \sqrt[3]{x} for when x=8.

Worked Solution
Create a strategy

Find the cube root of 8.

Apply the idea
\displaystyle \sqrt[3]{x}\displaystyle =\displaystyle \sqrt[3]{8}Substitute x=8
\displaystyle =\displaystyle 2Evaluate

Example 9

Write the following with a fractional index: \sqrt[7]{72}

Worked Solution
Create a strategy

Use the rule \sqrt[k]{b}=b^{\frac{1}{k}}.

Apply the idea

\sqrt[7]{72}=72^{\frac{1}{7}}

Idea summary

Negative exponent property: b^{-n}=\dfrac{1}{b^{n}}

Exponential unit fractions:

x^{\frac{1}{n}}=\sqrt[n]{x}

Outcomes

2.4.A

Rewrite exponential expressions in equivalent forms.

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