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Questions and Answers
What is an exponential function?
f(x) = b^x for b > 0, a ≠ 1
What is the natural exponential function?
f(x) = e^x
What is the value of the base e?
approximately 2.718281828
Which of the following is a law of exponents?
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The range of an exponential function is the set of all real numbers.
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What happens when you raise any base a to the power of 0?
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What equation can be used to solve a problem where both sides need to have the same base?
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Study Notes
Exponential Functions
- Exponential functions are mathematical functions of the form 𝑓(𝑥) = 𝑏 𝑥, where 𝑏 is a constant called the base and 𝑥 is a variable in the exponent.
- The base 𝑏 must be greater than 0 and not equal to 1.
- The domain of an exponential function is all real numbers.
- The range of an exponential function is all positive real numbers.
Base e Exponential Function
- The natural exponential function is defined as 𝑓(𝑥) = 𝑒 𝑥.
- The base 𝑒 is an irrational number approximately equal to 2.718281828.
Laws and Properties of Exponents
- Product of Powers: 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
- Quotient of Powers: 𝑎𝑚 / 𝑎𝑛 = 𝑎𝑚−𝑛
- Power of a Power: (𝑎𝑚)𝑛 = 𝑎𝑚∙𝑛
- Power of a Product: (𝑎𝑏)𝑚 = 𝑎𝑚 ∙ 𝑏 𝑚
- Power of a Quotient: (𝑎/𝑏)𝑛 = 𝑎𝑛 / 𝑏𝑛, where 𝑏 ≠ 0
- Negative Exponent: 𝑎−𝑛 = 1/𝑎𝑛
- Fractional Exponent: 𝑎𝑚/𝑛 = √𝑛𝑎𝑚
- Zero Exponent: 𝑎0 = 1
- Equality of Exponents: 𝑎 𝑥 = 𝑎 𝑦 if and only if x = y
- Equality of Bases: 𝑎 𝑥 = 𝑏 𝑥 if and only if a = b and for x ≠ 0
Example 7.1
- The example demonstrates how to simplify expressions with exponents using the laws of exponents.
- Part (a): 2a 3 4a 2 6a 5 = (2/4) * (a 3 / a 2) * (6 * a 5) = 3a 6
Example 7.2
- The example demonstrates how to solve exponential equations.
- Part (a): (1−x) 5 = (2x−1) 5. Since the bases are the same, we can set the exponents equal to each other. 1 - x = 2x - 1. Solving for x, we get x = 2/3.
Tutorial 7
- The tutorial provides practice problems for simplifying expressions and solving exponential equations.
- The problems require applying the laws of exponents and techniques for solving different types of exponential equations.
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Description
This quiz covers the fundamentals of exponential functions, including their definitions, properties, and the natural exponential function. Learn about the various laws of exponents and how they apply to different mathematical scenarios. Perfect for students looking to deepen their understanding of exponential mathematics.
