Problem 45 - Algebra and Trigonometry

Question

\(45-54\) . Use the Laws of Logarithms to combine the expression. $$ \log _{3} 5+5 \log _{3} 2 $$

Step-by-Step Solution

Answer

The combined expression is \(\log_3 160\).

Step 1: Apply Power Rule

The power rule of logarithms states that \( a \log_b c = \log_b c^a \). Here, we have \(5 \log_3 2\). We can apply the power rule to this expression as follows: \(5 \log_3 2 = \log_3 2^5\).

Step 2: Simplify Power Expression

Now calculate \(2^5\) to simplify the expression obtained in Step 1: \(2^5 = 32\). So, \(\log_3 2^5 = \log_3 32\).

Step 3: Apply Product Rule

After transforming the expression \(5 \log_3 2\) into \(\log_3 32\), we combine both logarithmic expressions using the product rule: \(\log_b m + \log_b n = \log_b (m \cdot n)\). Here, \(\log_3 5 + \log_3 32 = \log_3 (5 \times 32)\).

Step 4: Calculate Product and Simplify

Calculate the product inside the logarithm: \(5 \times 32 = 160\). Thus, the combined logarithm becomes \(\log_3 160\).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a handy tool for simplifying expressions where a logarithm is multiplied by a constant. Understanding this rule is essential in manipulating and combining logarithmic expressions efficiently. According to the power rule:

• \( a \log_b c = \log_b c^a \)

This means that if you have a constant multiplier next to a logarithm, you can move this constant as an exponent within the logarithm. This is particularly useful in breaking down complex logarithmic expressions into simpler forms that can be easily combined or further solved.
For instance, in the expression given in the problem, we had \(5 \log_3 2\). Using the power rule, we rewritten it as \(\log_3 2^5\), making it far easier to simplify the expression further. Instead of dealing with a multiple of a logarithm, we end up solving a single logarithm with an exponent. This method simplifies not just the expression, but often the process of problem-solving too, especially when combined with other logarithmic rules.

Product Rule of Logarithms

The product rule of logarithms is a core principle that helps in combining separate logarithmic expressions into a singular, streamlined form. This rule is exceptionally important when dealing with the sum of logarithms with the same base, making it possible to condense them into one.Here's how it works:

• \( \log_b m + \log_b n = \log_b (m \cdot n) \)

The rule essentially states that when you sum two logarithms that share the same base, they can be combined into a single logarithm, where their arguments (inside the logarithm) are multiplied together. This can really clear up messy expressions by removing unnecessary complexity and reducing the number of terms you have to handle.Continuing with the original exercise, after invoking the power rule, we had two expressions: \(\log_3 5\) and \(\log_3 32\). By applying the product rule, those could be combined into \(\log_3 (5 \times 32)\). This condensed our logarithmic expression, making the arithmetic operation easier to complete and understand.

Logarithmic Expressions

Logarithmic expressions can often seem daunting, but understanding how to manipulate them with the Laws of Logarithms simplifies many problems. The key to mastering logarithms lies in comprehending the various rules, like the power and product rules, to transform and combine expressions.A logarithmic expression might involve the sum or difference of multiple individual logs. Here are some tips to handle such expressions:

• Identify shared bases to apply rules like the power and product rules efficiently.
• Rewrite complicated products or quotients using equivalent logarithmic expressions.
• Use simplifications to turn products, powers, or divisions within logs into simpler forms.

The provided exercise is an example of utilizing these strategies, showcasing how a complex expression like \(\log_3 5 + 5\log_3 2\) can be transformed first by the power rule, and then condensed using the product rule. These expressions, initially featuring two separate and seemingly distinct parts, seamlessly combine into \(\log_3 160\). Understanding such transformations empowers students to tackle wider logarithmic problems with ease and confidence.