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Chapter 5: Problem 39

Express as a single logarithm and, if possible, simplify. $$\frac{1}{2} \log n+3 \log m$$

Short Answer

Expert verified
\( \log( \sqrt{n} \cdot m^3) \).

Step by step solution

01

Apply the power rule

Use the power rule of logarithms to move the coefficients in front of the logarithms as exponents. The power rule states that x \log_a b = \log_a b^x.So,\(\frac{1}{2} \log n\) becomes \(\log n^{1/2}\) and \(3 \log m\) becomes \(\log m^{3}\).
02

Rewrite the expression

Rewrite the expression with the exponents: \ \(\log n^{1/2} + \log m^{3}\).
03

Use the product rule

Apply the product rule of logarithms which states that\(\log_a b + \log_a c = \log_a(bc)\).Combine the logarithms into a single logarithm:\(\log(n^{1/2} \cdot m^3)\).
04

Simplify the final expression

Simplify the expression inside the logarithm if possible. Note that \(n^{1/2}\) can be written as \(\sqrt{n}\).Thus, the final expression is \( \log( \sqrt{n} \cdot m^3) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
When dealing with logarithms, the power rule is an essential tool. The power rule states that you can move a coefficient in front of the logarithm as an exponent within the logarithm. This is expressed as follows: \[ x \log_a b = \log_a b^x \]In our exercise, we apply the power rule to \[ \frac{1}{2} \log n \]This becomes \log n^{1/2}\, and to \3 \log m \, which becomes \log m^3\. Moving the coefficients as exponents helps in simplifying complex logarithmic expressions. This step is crucial before combining the terms using other logarithm rules such as the product rule.
Product Rule
The product rule is another fundamental property of logarithms. It states that the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments. Mathematically, this is written as: \[ \log_a b + \log_a c = \log_a (bc) \]Applying the product rule to our transformed expressions: \[ \log n^{1/2} + \log m^3 \]results in a single logarithm: \[ \log(n^{1/2} \cdot m^3) \]This rule significantly aids in reducing the complexity and number of logarithmic terms in expressions, making them easier to handle and understand.
Simplifying Logarithms
Once you have applied the power and product rules, the final step is to simplify the expression if possible. In our given exercise, the combined expression within the logarithm is: \[ \log(n^{1/2} \cdot m^3) \]Recognizing that \[ n^{1/2} \] is the same as \sqrt{n} \, we rewrite the expression, leading to a simplified form: \[ \log(\sqrt{n} \cdot m^3) \]This final step ensures the expression is in its most reduced and interpretable form, making it easier to work with in further calculations or applications.

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