Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
6. Exponential & Logarithmic Functions
Introduction to Logarithms
1:05 minutes
Problem 89
Textbook Question
Textbook QuestionIn Exercises 81–100, evaluate or simplify each expression without using a calculator. In e^6
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical expressions in the form of f(x) = a * b^x, where 'a' is a constant, 'b' is the base, and 'x' is the exponent. In this context, e^6 represents an exponential function where 'e' is the base of natural logarithms, approximately equal to 2.71828. Understanding the properties of exponential functions is crucial for evaluating expressions like e^6.
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The Number 'e'
The number 'e' is a fundamental constant in mathematics, known as Euler's number, and is the base of natural logarithms. It is an irrational number, meaning it cannot be expressed as a simple fraction, and it has important applications in calculus, particularly in growth and decay problems. Recognizing the significance of 'e' helps in understanding its role in exponential expressions.
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Simplifying Exponential Expressions
Simplifying exponential expressions involves applying the laws of exponents, which include rules such as a^m * a^n = a^(m+n) and (a^m)^n = a^(m*n). For the expression e^6, simplification may not be necessary, but understanding how to manipulate exponents is essential for more complex expressions. This knowledge allows for easier evaluation and comparison of exponential terms.
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