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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1:43 minutes

Problem 85

Textbook Question

In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C.logb 8

Verified step by step guidance

insert step 1> Recognize that 8 can be expressed as a power of 2, specifically 8 = 2^3.

insert step 2> Use the logarithmic identity: log_b(m^n) = n * log_b(m).

insert step 3> Apply the identity to the expression: log_b(8) = log_b(2^3).

insert step 4> Simplify using the identity: log_b(2^3) = 3 * log_b(2).

insert step 5> Substitute the given value: log_b(2) = A, so log_b(8) = 3A.

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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 (logb(mn) = logb(m) + logb(n)), the quotient rule (logb(m/n) = logb(m) - logb(n)), and the power rule (logb(m^k) = k * logb(m)). Understanding these properties is essential for rewriting logarithmic expressions in simpler forms.

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 logb(a) = logk(a) / logk(b) for any positive k. This is particularly useful when converting logarithmic expressions to a more manageable base, such as base 10 or base e.

Exponential Relationships

Logarithms are the inverse operations of exponentiation. For example, if logb(x) = y, then b^y = x. This relationship is crucial for understanding how to express logarithmic values in terms of their base and exponent, which is necessary for rewriting expressions like logb(8) in terms of logb(2) and logb(3).