Problem 71 - 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 \(C\). $$\log _{b} 8$$

Step-by-Step Solution

Answer

Hence, the expression \(\log_b 8\) in terms of \(A\) and \(C\) is \(3A\).

Step 1: Expression for \(\log_b 2\)

From the given, we have \(\log_b 2 = A\). This represents the logarithmic relation for '2' in base 'b' as 'A'.

Step 2: Expression for \(\log_b 8\)

We need to recall that \(8 = 2^3\). Hence, applying the logarithm on both sides, we get: \(\log_b 8 = \log_b (2^3)\). Using the property of logarithms, \(\log_b a^n = n \log_b a\), we have: \(\log_b 8 = 3 \log_b 2\).

Step 3: Substitute \(\log_b 2\) as 'A'

In the previous step, we obtained the expression \(\log_b 8 = 3 \log_b 2\). We already know that \(\log_b 2 = A\). Substituting this, we get: \(\log_b 8 = 3A\).