Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to rewrite and simplify the logarithmic expression.$$\log _{2} 4^{2} \cdot 3^{4}$$.

Step-by-Step Solution

Answer

The simplified expression is \( 4 + 4\log_{2}3 \).

Step 1: Apply the properties of logarithms

The expression is \( \log _{2} 4^{2} \cdot 3^{4} \). We can use the fact that logb(mn) = logb(m) + logb(n), so the expression can be rewritten as \( \log_{2}(4^2) + \log_{2}(3^4) \).

Step 2: Simplify the logarithms

We can then use the property that \(\log_{b}(m^n)\) = n ∗ \(\log_{b}(m)\) to further simplify. Thus, \(\log_{2}(4^2) = 2\log_{2}4\) and \(\log_{2}(3^4) = 4\log_{2}3\). The expression can be rewritten as \(2\log_{2}4 + 4\log_{2}3 \).

Step 3: Apply the change in base formula and simplify

The base 2 logarithm of 4 is 2, as \(2^2 = 4\). Thus, \( \log_{2}4 = 2\). Also, the base 2 logarithm of 3 reminds as is, as \( \log_{2}3 \). Consequently, \( 2 \log_{2}4 = 2*2 = 4 \) and \( 4 \log_{2}3 = 4 \log_{2}3\). The expression can then be rewritten as \( 4+ 4\log_{2}3 \).

Key Concepts

Properties of Logarithms

Logarithms have several important properties that make them useful for simplifying expressions. One key property is the Product Rule:

• For any positive numbers \( m \) and \( n \), and a base \( b \), \( \log_b(mn) = \log_b(m) + \log_b(n) \).

This property tells us that the logarithm of a product can be expressed as the sum of the logarithms.Another essential property is the Power Rule:

• For any positive number \( m \), a base \( b \), and an exponent \( n \), \( \log_b(m^n) = n \cdot \log_b(m) \).

This rule shows us that the logarithm of a power can be expressed as the product of the exponent and the logarithm of the base.Understanding these properties is crucial for simplifying logarithmic expressions efficiently. When faced with a complex expression, they allow us to break it down into simpler parts.

Logarithmic Expression

A logarithmic expression involves logarithms, such as \( \log_{b}x \), where \( b \) represents the base and \( x \) represents the argument. These expressions can often be complex due to multiple operations involving exponentiation and multiplication.When given a task to simplify a logarithmic expression such as \( \log _{2} 4^{2} \cdot 3^{4} \), our goal is to use the properties of logarithms to break the expression down into a simpler form.
This involves recognizing products within the expression and using the Product Rule and the Power Rule to rewrite and combine terms effectively.For example, \( \log_{2}(4^2 \cdot 3^4) \) can be initially rewritten by separating into two logarithmic terms: \( \log_{2}(4^2) + \log_{2}(3^4) \). This simplification process is central to managing and understanding logarithmic expressions.

Simplifying Logarithms

Simplifying logarithmic expressions involves using the properties of logarithms to reduce the expression to its simplest form. The process often includes several steps. First, apply the Product Rule to split a complex product into separate logarithmic terms. For instance, when simplifying \( \log_{2}(4^2 \cdot 3^4) \), we first break it into \( \log_{2}(4^2) + \log_{2}(3^4) \).Then, by using the Power Rule, these terms are simplified further. This involves transforming terms like \( \log_{2}(4^2) \) into \( 2\log_{2}(4) \), by pulling the exponent out as a multiplying factor.
Similarly, \( \log_{2}(3^4) \) becomes \( 4\log_{2}(3) \). Finally, whenever possible, calculate any known logarithmic values to reach a final simplistic form. In this example, we know \( \log_{2}4 \) is 2 because \( 2^2 = 4 \), allowing us to write \( 2\log_{2}4 \) as 4.Simplifying like this helps to analyze complex logarithmic expressions and makes them manageable.