Problem 12 - Elementary and Intermediate Algebra

Question

Match expression with an equivalent expression from the list on the right. \(\log _{3} \frac{10}{11}\) a. \(\frac{\log 11}{\log 3}\) b. \(11 \log _{3} 10\) c. \(\log _{3} 5+\log _{3} 2\) d. \(\log _{3} 10-\log _{3} 11\)

Step-by-Step Solution

Answer

The expression \(\log _{3} \frac{10}{11}\) matches with option d: \(\log _{3} 10 - \log _{3} 11\).

Step 1: Understand the original expression

The original expression is \(\log _{3} \frac{10}{11}\). This represents the logarithm to the base 3 of the fraction \(\frac{10}{11}\). We will find an equivalent expression from the provided options.

Step 2: Identify the appropriate logarithmic identity

Using the logarithmic property \(\log_{b} \frac{x}{y} = \log_{b} x - \log_{b} y\), we can expand the expression \(\log _{3} \frac{10}{11}\) as \(\log _{3} 10 - \log _{3} 11\).

Step 3: Match with given options

Now we compare our expanded form \(\log _{3} 10 - \log _{3} 11\) with the provided options. The correct match is option **d**, which is \(\log _{3} 10 - \log _{3} 11\).

Key Concepts

Logarithmic Identities

Logarithmic identities are a set of rules that help us manipulate and rewrite logarithmic expressions. They are especially useful when dealing with complex expressions, allowing us to simplify or rearrange them for easier computation. Here are some key logarithmic identities:

• The product rule: \( \log_{b}(xy) = \log_{b}x + \log_{b}y \)
• The quotient rule: \( \log_{b}\frac{x}{y} = \log_{b}x - \log_{b}y \)
• The power rule: \( \log_{b}(x^{c}) = c \cdot \log_{b}x \)

The exercise we looked at uses the quotient rule, which allows us to expand the logarithm of a fraction. In the original problem, recognizing that \( \log_{3}\frac{10}{11} \) can be expanded to \( \log_{3}10 - \log_{3}11 \) using this rule helped us find an equivalent expression.

Expressions

A mathematical expression is a combination of symbols and numbers arranged according to mathematical rules to represent a particular value. Logarithmic expressions specifically involve the logarithm function, which is essentially the inverse operation to exponentiation.In our exercise, the expression \( \log_{3}\frac{10}{11} \) is a logarithmic expression in its simplest form. Expressions can often be rewritten using identities or properties, providing us with different perspectives and enabling simpler calculations. It's essential to become comfortable with transforming expressions through identities in order to solve equations and evaluate values more quickly and accurately.

Equivalent Expressions

Equivalent expressions are different expressions that yield the same result for the same inputs. In the context of logarithms, using logarithmic identities helps us translate a complex or hard-to-solve expression into a form that might be easier to handle.In the exercise, we were tasked with identifying an expression equivalent to \( \log_{3}\frac{10}{11} \).

• By using the quotient rule, we transformed this into \( \log _{3} 10 - \log _{3} 11 \).
• This transformed expression matched option d. Despite appearing different from the original expression, both give the same number when calculated with the right value of 3 as the base.

Recognizing equivalent expressions allows for smoother problem solving, especially during transformations and simplifications in calculus and algebra.