Problem 7 - Algebra 2

Question

State the property or properties used to rewrite each expression. \(3 \log _{4} 5-3 \log _{4} 3=\log _{4}\left(\frac{5}{3}\right)^{3}\)

Step-by-Step Solution

Answer

The properties used in this case are the scaling property of logarithms (to move the '3' in front to the exponent position) and the subtraction property of logarithms (to combine the two separate logarithms into one log of a fraction).

Step 1: Identify the Relevant Properties of Logarithms

The properties of logarithms that are helpful in this case include the following two: 1. Scaling property: \(a \cdot \log_b c = \log_b (c^a)\)2. Subtraction property: \(\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)\)Here, 'a' is a constant and 'b' is the base of the logarithm.

Step 2: Apply the Scaling Property

We observe that both terms on the left-hand side of the equation have a common factor of '3'. With the scaling property, this can be rewritten as an exponent. Thus,\(3 \log_4 5 = \log_4 (5^3)\)\(3 \log_4 3 = \log_4 (3^3)\)

Step 3: Apply the Subtraction Property

The subtraction property tells us that subtracting two logarithms with the same base can be rewritten as a single logarithm, where the arguments divide each other. The left-hand side of the equation can thus be rewritten as \(\log_4 \left(\frac{5^3}{3^3}\right)\)

Key Concepts

Scaling Property of Logarithms

The scaling property of logarithms is a powerful tool that enables us to transform a multiplication situation inside a logarithmic context into an exponent. This is useful in simplifying expressions or solving logarithmic equations. The property is expressed as:

• \( a \cdot \log_b c = \log_b (c^a) \)

In this formula, 'a' is a multiplier that scales the logarithmic expression. The base 'b' remains constant. This property showcases one of the logarithm's main characteristics: converting multiplicative factors into exponents.
For instance, in the given problem, each term has a factor '3'. By applying the scaling property, you can transform terms like \(3 \log_4 5\) into \(\log_4 (5^3)\). Also, \(3 \log_4 3\) becomes \(\log_4 (3^3)\). Using this property can make calculations easier and simplify the manipulation of logarithmic expressions.

Subtraction Property of Logarithms

The subtraction property of logarithms is essential for combining logarithms into a single, more compact expression. It helps handle expressions where you see a difference between two logarithms. The property is given by:

• \( \log_b a - \log_b c = \log_b \left(\frac{a}{c}\right) \)

Here, both logarithmic terms must have the same base 'b'. This property effectively turns the subtraction of separate logarithmic expressions into a division inside a single logarithm.
In the problem at hand, after applying the scaling property, you have logarithms with matching bases. Therefore, you can simplify the difference of the logs, \(\log_4 (5^3) - \log_4 (3^3)\), to a single logarithmic term, \(\log_4 \left(\frac{5^3}{3^3}\right)\). This technique is crucial when simplifying complex logarithmic equations and can make further calculations much more straightforward.

Exponents in Logarithmic Expressions

Exponents frequently appear in logarithmic expressions, and understanding how to manipulate them is vital for simplifying equations and expressions. When exponents appear, using properties of logarithms can help manage them effectively.

• In the scaling property, exponents arise naturally by transforming a multiplication factor into an exponent.
• The expression \(a \cdot \log_b c\) becomes \(\log_b (c^a)\), indicating that the number 'a' is now an exponent.

This reflects the concept that logarithms are the inverses of exponents. By applying the properties of logarithms, you gain a powerful toolbox for tackling exponents in these expressions.
In our example, transforming the expression \(3 \log_4 5 - 3 \log_4 3\), after using the scaling property, incorporates exponents as in \(\log_4 (5^3)\) and \(\log_4 (3^3)\). Ultimately, with the subtraction property, they simplify to \(\log_4 \left(\frac{5^3}{3^3}\right)\), highlighting how exponents get managed gracefully in logarithmic contexts. This understanding can crucially aid in solving various problems involving logarithms and exponents efficiently.