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
3:04 minutes
Problem 85a
Textbook Question
Textbook QuestionSolve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2 log x−log 7=log 112
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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 (log_b(MN) = log_b(M) + log_b(N)), the quotient rule (log_b(M/N) = log_b(M) - log_b(N)), and the power rule (log_b(M^p) = p * log_b(M)). These properties allow us to manipulate logarithmic expressions to isolate the variable.
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Domain of Logarithmic Functions
The domain of a logarithmic function is restricted to positive real numbers. This means that any argument of a logarithm must be greater than zero. When solving logarithmic equations, it is crucial to check the solutions against the original equation to ensure they fall within this domain, as extraneous solutions may arise from the manipulation of logarithmic expressions.
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Graphs of Logarithmic Functions
Decimal Approximation
After finding the exact solution to a logarithmic equation, it may be necessary to provide a decimal approximation. This involves using a calculator to evaluate logarithmic expressions to a specified number of decimal places. For instance, if the exact solution is log_x(112) = 2, the decimal approximation would be calculated to two decimal places, providing a more intuitive understanding of the solution.
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