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Evaluate or simplify each expression without using a calculator. In 1
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 85
In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C.
logb 8
Guida verificata passo dopo passo1
Recognize that the expression log_b 8 can be rewritten using the fact that 8 is a power of 2. Since 8 = 2^3, rewrite the logarithm as log_b (2^3).
Apply the logarithmic power rule, which states that log_b (x^n) = n * log_b x. Using this, rewrite log_b (2^3) as 3 * log_b 2.
Recall the given information that log_b 2 = A. Substitute this into the expression to get 3 * A.
Thus, the expression log_b 8 is written in terms of A as 3A.
No further simplification is needed since the problem asks to express the logarithm in terms of A and C.
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Logarithm Properties
Logarithms have specific properties such as the product, quotient, and power rules. The power rule states that log_b(x^n) = n * log_b(x), which allows us to simplify expressions like log_b(8) by expressing 8 as a power of a base number.
Video consigliato:
5:36
Change of Base Property
Change of Base and Given Logarithms
When given specific logarithmic values like log_b(2) = A and log_b(3) = C, we can rewrite other logarithms in terms of these known values by expressing the argument as products or powers of 2 and 3. This helps in expressing complex logs in simpler terms.
Video consigliato:
5:36Change of Base Property
Prime Factorization
Prime factorization involves expressing a number as a product of prime numbers. For example, 8 can be written as 2^3. This factorization is essential to rewrite log_b(8) in terms of log_b(2), which is given as A, enabling simplification using logarithm properties.
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07:30Introduction to Factoring Polynomials
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