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

Verified step by step guidance

1. 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).
2. 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.
3. Recall the given information that log_b 2 = A. Substitute this into the expression to get 3 * A.
4. Thus, the expression log_b 8 is written in terms of A as 3A.
5. No further simplification is needed since the problem asks to express the logarithm in terms of A and C.

Key Concepts

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.

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.

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.