Problem 72 - Introductory and Intermediate Algebra for College Students 4th

Question

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). $$\log _{b} 81$$

Step-by-Step Solution

Answer

The expression \( \log _{b} 81 \) can be written as \( 4C \) in terms of given logarithms.

Step 1: Express 81 as product or power of known values

Start by exploring the factorization of the number 81. As we can see, \( 81 = 3^4 \). It will help us to express \( \log _{b} 81 \) in terms of C.

Step 2: Use Logarithm power rule

The power rule of logarithms allows us to transfer the exponent of the argument of the logarithm in front of the logarithm. This rule can be expressed as \( \log_b m^n = n \log_b m \). Exploiting this rule, we express \( \log _{b} 81 \) or \( \log _{b} 3^4 \) as \( 4 \log _{b} 3 \)

Step 3: Substitute C for log_b 3

As per the given problem, we know \( \log_b 3 = C \) . Substituting C in place of \( \log _{b} 3 \) in above derived equation, we get this as \( 4 * C \)