Problem 24 - College Algebra

Question

For the following exercises, condense each expression to a single logarithm using the properties of logarithms. $$ 4 \log _{7}(c)+\frac{\log _{7}(a)}{3}+\frac{\log _{7}(b)}{3} $$

Step-by-Step Solution

Answer

\( \log_{7}(c^4 \cdot a^{1/3} \cdot b^{1/3}) \)

Step 1: Apply Power Rule

The power rule for logarithms states that \(k \log_b(x) = \log_b(x^k)\). Apply this rule to each logarithmic term: 1. \(4 \log_{7}(c) = \log_{7}(c^4)\)2. \(\frac{\log_{7}(a)}{3} = \log_{7}(a^{1/3})\)3. \(\frac{\log_{7}(b)}{3} = \log_{7}(b^{1/3})\)

Step 2: Apply Product Rule

The product rule for logarithms states that \(\log_b(x) + \log_b(y) = \log_b(xy)\). Add the terms obtained in Step 1 using this rule:\[ \log_{7}(c^4) + \log_{7}(a^{1/3}) + \log_{7}(b^{1/3}) = \log_{7}(c^4 \cdot a^{1/3} \cdot b^{1/3}) \]

Step 3: Write the Final Expression

Combine the expressions from Step 2 to write the condensed logarithm:\[ \log_{7}(c^4 \cdot a^{1/3} \cdot b^{1/3}) \]

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a fundamental concept that simplifies expressions involving logarithms. It states that for any real number \(k\) and positive number \(x\), and a logarithm with base \(b\), you can rewrite \(k \log_b(x)\) as \(\log_b(x^k)\). This rule comes in handy when dealing with coefficients in front of logarithms because it allows you to "bring the power up" next to the argument of the logarithm.
For example, in the expression \(4 \log_7(c)\), using the power rule, you convert it to \(\log_7(c^4)\). Similarly, if you have a fractional coefficient, like \(\frac{1}{3} \log_7(a)\), it becomes \(\log_7(a^{1/3})\) upon applying the power rule. This transformation enables you to manage different terms more easily in various logarithmic expressions.
Understanding this rule is fundamental when you need to work with compound logarithms because it helps simplify the expression by combining similar terms.

Product Rule of Logarithms

The product rule of logarithms is another essential property used when condensing expressions into a single logarithm. It states that when you have a sum of two logarithms with the same base, \(\log_b(x) + \log_b(y)\), it can be rewritten as a single logarithm, \(\log_b(xy)\). This is particularly useful when trying to simplify and condense logarithmic expressions.
For instance, suppose you have a transformed expression like \(\log_7(c^4) + \log_7(a^{1/3}) + \log_7(b^{1/3})\). By applying the product rule, you merge them into a single logarithmic term: \(\log_7(c^4 \cdot a^{1/3} \cdot b^{1/3})\). This step reduces complexity, making it easier to evaluate or manipulate the expression further.

• The product rule essentially helps combine multiple logarithmic terms into one unified expression.
• It's very useful for solving or simplifying equations involving logarithms.

This rule holds only when the base of all logarithmic terms is the same, which is why it's crucial to first verify the base before applying the rule.

Condensing Logarithmic Expressions

Condensing logarithmic expressions involves transforming a complex series of logarithmic terms into a single, more streamlined logarithm. This process typically uses the power rule and product rule we've discussed.
The goal is to take multiple logarithms in a sum or difference and simplify them into one expression. When given an expression like \(4 \log _{7}(c)+\frac{\log _{7}(a)}{3}+\frac{\log _{7}(b)}{3}\), you apply the logarithm properties:

• First, use the power rule to adjust coefficients, turning them into exponents.
• Next, use the product rule to combine terms if they are being added.

The condensed form is more efficient and is often necessary for advanced problem-solving, especially when working with equations or modeling scenarios involving logarithms. This process not only makes the expression easier to work with but also reveals insights about the relationships between the different components within the problem.
Always remember that the purpose of condensing is to simplify and unify the expression while maintaining its mathematical integrity.