Problem 24 - Algebra 2

Question

Expand each logarithm. \(\log _{3}(2 x)^{2}\)

Step-by-Step Solution

Answer

The expanded form of \(\log _{3}(2 x)^{2}\) is \(2*\log_{3}2 + 2*\log_{3}x\).

Step 1: Apply the power rule of logarithms

The given expression is \(\log_{3}(2x)^{2}\). Apply the power rule of logarithms to move the exponent (2) in front of the logarithm: \(2*\log_{3}(2x)\).

Step 2: Expand the log expression

Now it is time to apply the product rule of logarithms which states that \(log_{b}(m*n) = log_{b}(m) + log_{b}(n)\). Apply this rule to \(\log_{3}(2x)\) which will give \(2*[\log_{3}2 + \log_{3}x]\).

Step 3: Distribute the coefficient in the expression

Distribute the '2' inside the brackets, it will give \(2*\log_{3}2 + 2*\log_{3}x\). This is the final expanded form of the given logarithmic expression.

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a handy tool for simplifying expressions with exponents inside logarithms. It states that for any positive number \( a \), and a real number \( b \), the logarithm of \( a^b \), denoted \( \log_b(a^c) \), can be rewritten as \( c \times \log_b(a) \). This rule allows us to "bring down" the exponent to become a multiplier of the log, simplifying our calculations.
For instance, in the expression \( \log_{3}(2x)^2 \), we use the power rule to transform it into \( 2 \times \log_{3}(2x) \). This conversion simplifies the log expression, making it easier to work with.
Remembering this handy rule can prevent errors and significantly streamline solving complex logarithmic expressions. Always look for opportunities to apply the power rule when dealing with exponents inside a logarithm.

Product Rule of Logarithms

The product rule of logarithms helps us break down complex logarithmic expressions into simpler parts. According to this rule, the logarithm of a product \( m \times n \) can be expressed as a sum: \( \log_b(m \times n) = \log_b(m) + \log_b(n) \). This is incredibly useful when dealing with products inside logarithms, allowing us to separate terms and simplify our problem.
For example, after applying the power rule to \( \log_{3}(2x)^2 \), we need to handle \( \log_{3}(2x) \). Using the product rule here, we convert it to \( \log_{3}(2) + \log_{3}(x) \).

• The product of elements inside the logarithm becomes a sum of individual logarithms.
• This conversion is essential for the full expansion of logarithmic expressions.

Breaking down expressions using the product rule can simplify calculations and make the problem more manageable.

Expand Logarithms

Expanding logarithms is the process of expressing a logarithm as a sum or difference of simpler logarithms. This makes it easier to solve or simplify logarithmic equations. By combining the power and product rules, we can expand even complex expressions.

Let's examine the problem \( \log_{3}(2x)^2 \). We've already applied the power rule to get \( 2 \times \log_{3}(2x) \). Next, by using the product rule, we transform \( \log_{3}(2x) \) into \( \log_{3}2 + \log_{3}x \). Now, our expression looks like this: \( 2 \times (\log_{3}2 + \log_{3}x) \).
Finally, distribute the multiplier \( 2 \) throughout the expanded terms:

• \(2 \times \log_{3}2 \)
• \(+ 2 \times \log_{3}x \)

This distribution yields \( 2 \times \log_{3}2 + 2 \times \log_{3}x \).
Expanding logarithms allows you to break down expressions into more manageable components, facilitating ease of use and clarity in solving logarithmic problems.