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)

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Recall the logarithm property for division:

\log b (\frac{3}{2}) = \log b 3 - \log b 2

. This means the log of a quotient is the difference of the logs.

Substitute the given values: since

\log b 2 = A

and

\log b 3 = C

, replace these in the expression.

Write the expression as

\log b (\frac{3}{2}) = C - A

This expresses

\log b (\frac{3}{2})

entirely in terms of the variables A and C as requested.

No further simplification is needed since the problem asks only to write the expression in terms of A and C.

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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.

In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C.logb (3/2)