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**Finding Equations of Exponential Function**

Section 4.4 Finding Equations of Exponential Function

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**Finding an Equation of an Exponential Curve**

Using the Base Multiplier Property to Find Exponential Functions Example An exponential curve contains the points listed in the table. Find an equation of the curve. Solution Exponential is of the form f(x) = abx y-intercept is (0, 3), so a = 3 Input increases by 1, output multiplies by 2: b = 2 f(x) = 3(2)x Section 4.4 Slide 2

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**Verify results using graphing calculator**

Finding an Equation of an Exponential Curve Using the Base Multiplier Property to Find Exponential Functions Solution Continued Verify results using graphing calculator Section 4.4 Slide 3

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**Linear versus Exponential Functions**

Using the Base Multiplier Property to Find Exponential Functions Example Find a possible equation of a function whose input – output pairs are listed in the table. Solution x increases by 1, y multiplies by 1/3: b = 1/3 y-intercept is (0, 162): a = 162 . Section 4.4 Slide 4

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**Linear versus Exponential Functions**

Using the Base Multiplier Property to Find Exponential Functions Example 2. Find a possible equation of a function whose input – output pairs are listed in the table. Solution x increases by 1, y subtracted by 4: Linear function y-intercept is (0, 50) y = 4x + 50 Section 4.4 Slide 5

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**Find all real-number solutions.**

Linear versus Exponential Functions Solving Equations of the Form abn = k for b Example Find all real-number solutions. Solution 1. Solutions are 5 and –5 Use the notation 5 Section 4.4 Slide 6

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**2. 3. Check that both –2 and 2 satisfy the equation.**

Linear versus Exponential Functions Solving Equations of the Form abn = k for b Solution 2. 3. Check that both –2 and 2 satisfy the equation. Section 4.4 Slide 7

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**Linear versus Exponential Functions**

Solving Equations of the Form abn = k for b Solution 4. Check that 1.55 approx. satisfies the equation. 5. The equation b6 = –28 has no real solution, since an even exponent gives a positive number. Section 4.4 Slide 8

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**Solving Equations of the Form bn = k for b**

Solving Equations of the Form abn = k for b Summary To solve an equation of the form bn = k for b, If n is odd, the real-number solution is If n is even, and k ≥ 0, the real-number solutions are If n is even and k < 0, there is no real number solution. Section 4.4 Slide 9

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**One-Variable Equations Involving Exponents**

Solving Equations of the Form abn = k for b Example Find all real-number solutions. Round your answer to the second decimal place. 5.42b6 – 3.19 = Solution Section 4.4 Slide 10

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**One-Variable Equations Involving Exponents**

Solving Equations of the Form abn = k for b Solution Continued 2. Section 4.4 Slide 11

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**Finding Equations of an Exponential Function**

Using Two Points to Find Equations of Exponential Function Example Find an approximate equation y = abx of the exponential curve that contains the points (0, 3) and (4, 70). Round the value of b to two decimal places. y-intercept is (0, 3): y = 3bx Substitute (4, 70) and solve for b Solution Section 4.4 Slide 12

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**Finding Equations of an Exponential Function**

Using Two Points to Find Equations of Exponential Function Solution Continued Our equation is y = 3(2.20)x Graph contains (0, 3) b is rounded Doesn’t go through (0, 70), but it’s close Section 4.4 Slide 13

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Natural Logarithms Section 5.6. Lehmann, Intermediate Algebra, 4ed Section 5.6Slide 2 Definition of Natural Logarithm Definition: Natural Logarithm A.

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