Problem 24 - College Algebra with Corequisite Support

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

The expression condenses to \( \log_7(c^4 \cdot a^{1/3} \cdot b^{1/3}) \).

Step 1: Apply the Power Rule of Logarithms

The power rule of logarithms states that when you have a term of the form \( k \log_b(x) \), it can be rewritten as \( \log_b(x^k) \). Using this rule, rewrite the expression using powers for each logarithmic term. Thus, \(4 \log_7(c)\) becomes \(\log_7(c^4)\), \(\frac{\log_7(a)}{3}\) becomes \(\log_7(a^{1/3})\), and \(\frac{\log_7(b)}{3}\) becomes \(\log_7(b^{1/3})\).

Step 2: Use the Product Rule of Logarithms

The product rule of logarithms says \( \log_b(M) + \log_b(N) = \log_b(MN) \). Applying this rule to our expression, begin by combining the first two logarithms: \( \log_7(c^4) + \log_7(a^{1/3}) = \log_7(c^4 \cdot a^{1/3}) \). Then combine this result with the third term: \( \log_7(c^4 \cdot a^{1/3}) + \log_7(b^{1/3}) = \log_7(c^4 \cdot a^{1/3} \cdot b^{1/3}) \).

Step 3: Simplify the Expression

Now simplify the expression into a single logarithm: \( \log_7(c^4 \cdot a^{1/3} \cdot b^{1/3}) \). The entire expression is now condensed into one logarithmic term with base 7, reflecting the product of \( c^4 \), \( a^{1/3} \), and \( b^{1/3} \).

Key Concepts

Power Rule of Logarithms

In mathematics, logarithms have various properties that make them quite useful, especially for simplifying complex expressions. One key property is the power rule. This rule tells us that if you have a logarithm in the form of \( k \log_b(x) \), it can be rewritten as \( \log_b(x^k) \). This is particularly helpful when dealing with multiplication within logarithms.

For example, in our given exercise, we have \( 4 \log_7(c) \). Using the power rule, we transform it to \( \log_7(c^4) \). Similarly, \( \frac{\log_7(a)}{3} \) and \( \frac{\log_7(b)}{3} \) become \( \log_7(a^{1/3}) \) and \( \log_7(b^{1/3}) \), respectively.

This conversion allows for easier manipulation of the terms, which is useful when you're preparing to use further logarithmic properties, such as condensing.

Product Rule of Logarithms

Another essential property of logarithms is the product rule. This rule tremendously helps in combining logarithmic terms. It states that the sum of two logarithms with the same base can be combined into a single logarithm, expressed as \( \log_b(M) + \log_b(N) = \log_b(MN) \). This rule transforms addition within logarithms into multiplication.

In our exercise, after applying the power rule, we have terms like \( \log_7(c^4) \), \( \log_7(a^{1/3}) \), and \( \log_7(b^{1/3}) \). By applying the product rule, the expression \( \log_7(c^4) + \log_7(a^{1/3}) \) becomes \( \log_7(c^4 \cdot a^{1/3}) \).

Further, including the third term gives us \( \log_7(c^4 \cdot a^{1/3} \cdot b^{1/3}) \). By utilizing the product rule, what began as multiple separate logarithmic terms are now compactly expressed as one logarithm.

Condensing Logarithms

Condensing logarithms refers to the process of combining multiple logarithmic expressions into a single term. This is accomplished by using various logarithmic properties, including the power and product rules.

To condense a logarithm, first, apply the power rule to address any coefficients. Then, use the product rule to combine the resulting terms. This process aligns with our exercise:

• Transform \( 4 \log_7(c) \) into \( \log_7(c^4) \), \( \frac{\log_7(a)}{3} \) into \( \log_7(a^{1/3}) \), and \( \frac{\log_7(b)}{3} \) into \( \log_7(b^{1/3}) \).
• Apply the product rule to get a single term: \( \log_7(c^4 \cdot a^{1/3} \cdot b^{1/3}) \).

Now, the entire logarithm is simplified and condensed. Condensing logarithms is a valuable skill in mathematics, offering more streamlined expressions for easier interpretation and manipulation.