Problem 12 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{5}\left(\frac{125}{y}\right) $$

Step-by-Step Solution

Answer

The expanded form of \(\log _{5}\left(\frac{125}{y}\right)\) is \(3 - \log_{5}{y}\)

Step 1: Apply the Quotient Rule

Use the quotient rule \(\log_b{\frac{A}{B}} = \log_b{A} - \log_b{B}\) to divide the log into two. This gives: \(\log _{5}\left(\frac{125}{y}\right) = \log_{5}{125} - \log_{5}{y}\)

Step 2: Apply the Power Rule

Note that \(125 = 5^{3}\). Therefore, \(\log_{5}{125}\) can be written as \( \log_{5}{5^{3}}\). Using the power rule, this can be simplified to \(3 \cdot \log_{5}{5}\). This gives: \(\log _{5}\left(\frac{125}{y}\right)= 3 \cdot \log_{5}{5} - \log_{5}{y}\)

Step 3: Evaluate the Expressions

Note that \(\log_{b}{b} = 1\). Therefore, \( \log_{5}{5} = 1\), which means that \(3 \cdot \log_{5}{5} = 3 \cdot 1 = 3\). Substitute the value back into the expression, giving \(3 - \log_{5}{y}\)

Key Concepts

Quotient Rule

The Quotient Rule is an essential property of logarithms that helps in simplifying complex logarithmic expressions. This rule states that the logarithm of a quotient is equivalent to the difference of the logarithms of the numerator and the denominator. In mathematical terms, it is expressed as:

• \(\log_b{\frac{A}{B}} = \log_b{A} - \log_b{B}\)

Using this rule makes it easier to break down and manipulate log expressions that involve division. For example, if you have \(\log_5{\left(\frac{125}{y}\right)}\), you can simplify it as \(\log_5{125} - \log_5{y}\). Applying the quotient rule clarifies the computation by separating it into simpler components.

Power Rule

The Power Rule of logarithms is another pivotal property that aids in simplifying expressions. It states that the logarithm of a power can be simplified by multiplying the exponent in front of the log. The power rule is expressed mathematically as:

• \(\log_b{(A^n)} = n \cdot \log_b{A}\)

This rule is handy when dealing with numbers that can be expressed as powers of the base. For instance, if we recognize \(125\) as \(5^3\), then \(\log_5{125}\) becomes \(\log_5{5^3}\), which simplifies to \(3 \cdot \log_5{5}\) using the power rule. It turns complicated calculations into more straightforward problems.

Logarithmic Expressions

Logarithmic expressions often involve complex terms and require understanding various properties to simplify them. Breaking down such expressions into simpler parts using properties of logs, like the quotient and power rules, helps manage them more efficiently.

• The quotient rule is used to separate log expressions involving division.
• The power rule simplifies expressions with exponents.

By applying these rules systematically, complex expressions like \(\log_5{\left(\frac{125}{y}\right)}\) are turned into easier components such as \(3 \cdot \log_5{5} - \log_5{y}\). Thus, mastering these core properties allows for effective manipulation and simplification of log expressions.

Evaluating Logarithms

Evaluating logarithms involves calculating the value of log expressions without relying on a calculator. It often requires applying properties like the basic definition, where \(\log_b{b} = 1\), and other rules such as the quotient and power rules.

• For base cases, \(\log_b{b} = 1\), which helps in direct evaluations.
• For expressions like \(3 \cdot \log_5{5}\), knowing \(\log_5{5} = 1\) simplifies calculations as it equals \(3\cdot1=3\).

These methods lead to more efficient problem-solving as they sharpen your ability to evaluate logs manually, consolidating your understanding of logarithmic properties and their applications.