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 (3/2)

Verified step by step guidance

1. Recall the logarithm property for division: $\log b_{}\left(\frac{3}{2}\right)=\log b_{}3-\log b_{}2$. This means the log of a quotient is the difference of the logs.
2. Substitute the given values: since $\log b_{}2=\; A$ and $\log b_{}3=\; C$, replace these in the expression.
3. Write the expression as $\log b_{}\left(\frac{3}{2}\right)=\; C\; -\; A$.
4. This expresses $\log b_{}\left(\frac{3}{2}\right)$ entirely in terms of the variables A and C as requested.
5. No further simplification is needed since the problem asks only to write the expression in terms of A and C.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the quotient rule: log_b(x/y) = log_b(x) - log_b(y). This allows us to rewrite complex logarithmic expressions as differences or sums of simpler logs.

Change of Base and Logarithm Notation

Understanding the notation log_b(x) means the logarithm of x with base b is crucial. Given log_b(2) = A and log_b(3) = C, we can express other logarithms with base b in terms of A and C by applying logarithmic properties.

Expressing Logarithmic Expressions in Terms of Variables

When given variables representing logarithms, such as A and C, the goal is to rewrite expressions like log_b(3/2) using these variables. This involves substituting and simplifying using known values and logarithmic rules.