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
Solving Exponential and Logarithmic Equations
5:13 minutes
Problem 65c
Textbook Question
Textbook QuestionSolve each equation. Give solutions in exact form. See Examples 5–9. log_8 (x + 2) + log_8 (x + 4) = log_8 8
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Properties
Understanding the properties of logarithms is essential for solving logarithmic equations. Key properties include the product rule, which states that log_b(m) + log_b(n) = log_b(m*n), and the fact that log_b(b) = 1. These properties allow us to combine or simplify logarithmic expressions, making it easier to isolate the variable.
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Change of Base Property
Change of Base Formula
The change of base formula is useful when dealing with logarithms of different bases. It states that log_b(a) can be expressed as log_k(a) / log_k(b) for any positive k. This concept is particularly helpful when the base of the logarithm is not easily manageable, allowing for conversion to a more familiar base, such as 10 or e.
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Change of Base Property
Exponential Equations
Solving logarithmic equations often involves converting them into exponential form. For example, if log_b(a) = c, then a = b^c. This understanding is crucial for finding the values of x in logarithmic equations, as it allows us to express the logarithmic equation in a more straightforward algebraic form that can be solved for the variable.
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Solving Exponential Equations Using Logs
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