Table of contents

0. Review of Algebra4h 18m
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 22m
10. Combinatorics & Probability1h 45m

6. Exponential & Logarithmic Functions

Properties of Logarithms

College Algebra

6. Exponential & Logarithmic Functions

Properties of Logarithms

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2:02 minutes

Problem 130

Textbook Question

If log 3 = A and log 7 = B, find log7 (9) in terms of A and B.

Verified step by step guidance

Start by expressing $ \log_7(9) $ using the change of base formula: $ \log_7(9) = \frac{\log(9)}{\log(7)} $.

Recognize that $ \log(9) $ can be rewritten using the property of logarithms: $ \log(9) = \log(3^2) = 2\log(3) $.

Substitute $ \log(3) = A $ into the expression: $ \log(9) = 2A $.

Substitute $ \log(7) = B $ into the change of base formula: $ \log_7(9) = \frac{2A}{B} $.

Conclude that $ \log_7(9) $ in terms of $ A $ and $ B $ is $ \frac{2A}{B} $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Properties

Logarithmic properties are rules that govern the manipulation of logarithms. 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^k) = k * log_b(m)). Understanding these properties is essential for simplifying logarithmic expressions and solving equations involving logarithms.

Change of Base Formula

The change of base formula allows us to express logarithms in terms of logarithms of a different base. It states that log_b(a) = log_k(a) / log_k(b) for any positive k. This formula is particularly useful when the logarithm's base is not easily computable, enabling us to convert it into a more manageable form using known logarithmic values.

Expressing Logarithms in Terms of Known Values

Expressing logarithms in terms of known values involves using given logarithmic values to rewrite other logarithmic expressions. In this case, we can use the values of log 3 = A and log 7 = B to find log 9 in terms of A and B. This technique often involves applying logarithmic properties and the change of base formula to derive the desired expression.