Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions. $$\log _{3} 405-\log _{3} 5$$

Step-by-Step Solution

Answer

The expression \(\log _{3} 405-\log _{3} 5\) simplifies to \(4\).

Step 1: Identify the Property to Use

Recognise that the problem involves the difference of two logarithms with the same base. In such scenarios, the quotient rule for logarithms applies.

Step 2: Use the Quotient Rule for Logarithms

According to the quotient rule, \(\log_{b} (p) - \log_{b} (q) = \log_{b} (\frac{p}{q})\). Applying this rule to our expression, we get \(\log _{3} 405-\log _{3} 5 = \log_{3} (\frac{405}{5})\).

Step 3: Simplify the Fraction

The fraction inside the parenthesis simplifies to \(81\). So, we get \(\log_{3} (\frac{405}{5}) = \log_{3} (81)\).

Step 4: Evaluate the Logarithmic Expression

The logarithm base 3 of 81 equals 4 because \(3^4=81\). So, the whole expression simplifies as: \(\log_{3} (81)=4\).

Key Concepts

Quotient Rule for Logarithms

When dealing with the difference between two logarithms that share the same base, the quotient rule for logarithms becomes incredibly useful. This rule helps to combine these two logarithms into a single expression. The rule states that if you have \( \log_b(p) - \log_b(q) \), it can be rewritten as \( \log_b\left(\frac{p}{q}\right) \). This property stems from the idea that subtraction of logarithms represents division of the original numbers.
By applying this rule, complex logarithmic expressions can often be greatly simplified, making them easier to evaluate.

Logarithmic Expressions

A logarithmic expression involves one or more logarithms. Logarithmic expressions help us represent exponential relationships in a different form. They come in handy especially when solving for exponents or managing complex calculations.
Just like regular algebraic expressions, logarithmic expressions can be simplified using properties of logarithms: the product rule, power rule, and the quotient rule. In problems like the one presented, simplifying logarithmic expressions using these properties often leads to much simpler calculations.

Simplifying Fractions

In the context of logarithmic expressions, simplifying fractions plays a crucial role, as seen when applying the quotient rule. After using the quotient rule to condense the logarithms, it's common to encounter a fraction within the logarithmic term.
To simplify such fractions, you need to divide the numerator by the denominator, if possible. For example, in our exercise, after using the quotient rule, we end up with \( \frac{405}{5} \), which simplifies neatly to 81. Simplifying fractions effectively reduces complexity and prepares the expression for step-by-step evaluation.

Evaluating Logarithms

Evaluating a logarithm involves calculating the power to which the base of the logarithm must be raised to obtain a particular number. In the context of our exercise, we evaluate the logarithm \( \log_3(81) \).
To solve this, you ask: "What power should three be raised to, in order to yield 81?" The answer is 4, because \( 3^4 = 81 \). Evaluating logarithms is the final step in simplifying logarithmic expressions, helping you reach a numerical conclusion.