Problem 13 - Algebra 2

Question

Write each logarithmic expression as a single logarithm. \(5 \log 3+\log 4\)

Step-by-Step Solution

Answer

The expression \(5 \log 3+\log 4\) simplifies to \(\log 972\).

Step 1: Apply the power rule

Given that \(5 \log 3+\log 4\), and using the \(\log (x^n) = n \log x\) rule in reverse, rewrite the \(5 \log 3\) as \(\log 3^5\), so the initial expression becomes \(\log 3^5+\log 4\)

Step 2: Use the product rule

The next step involves using the \(\log(xy)=\log x+\log y\) rule. The expression can be rewritten as a single log of the products, \(3^5\) and 4, making the equation \(\log 3^5 \cdot 4\).

Step 3: Simplify

Perform the multiplication inside the log to obtain the final expression. This results in the equation \(\log (243 \cdot 4)\). The multiplication gives the final result as \(\log 972\).

Key Concepts

Understanding the Power Rule in Logarithms

The power rule is a fundamental aspect of simplifying logarithmic expressions. This rule states that for any logarithm of the form \( log(x^n) = n \, log(x) \). Here, the exponent \( n \) can be brought in front of the logarithm as a coefficient. It is a handy tool because it allows us to move between forms with exponents and simpler coefficients.

• In the given exercise, \( 5 \, log(3) \) becomes \( log(3^5) \) when we apply the power rule.
• The exponent, 5, is moved from being a coefficient of \( log(3) \) to an exponent of 3.

This transformation helps in the further process of combining and simplifying logarithmic expressions. The power rule is especially useful when you need to reverse the operation, such as when dealing with coefficients in front of logs.

The Product Rule in Logarithms Simplification

The product rule for logarithms is another crucial concept to master when simplifying expressions. According to this rule, the sum of two logarithms can be combined into a single logarithm: \( log(x) + log(y) = log(xy) \).

• In our exercise, after applying the power rule, the expression is \( log(3^5) + log(4) \).
• The product rule simplifies this to \( log(3^5 \times 4) \) or \( log(972) \) after calculation.

Using the product rule, we can effectively condense multiple logarithmic terms into one, making evaluations simpler and more straightforward. This rule is particularly powerful when you encounter a sum of logarithms and need to express them compactly.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions often involves applying rules like the power and product rules. This primary goal is to convert multiple log terms into a single, simplified expression.

• In the exercise provided, simplification led us from initial separate terms to the single expression \( log(972) \).
• Simplification requires rewriting terms using rules and performing necessary arithmetic operations.

Understanding how to simplify logarithmic expressions is an essential skill because it conveys the original information in a more concise manner. It's especially helpful in solving equations, where simpler forms lead to easier computations.