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Chapter 9: Problem 30

Express as an equivalent expression that is a single logarithm. $$\log _{a} 26-\log _{a} 2$$

Short Answer

Expert verified
\log_a 13

Step by step solution

01

- Understand the Logarithm properties

Recall that \( \log_a b - \log_a c = \log_a (\frac{b}{c}) \). This property will help solve the problem.
02

- Apply the property

Using the property from Step 1, apply it to the given expression: \( \log_a 26 - \log_a 2 \)
03

- Simplify the expression

Combine the logarithms into a single logarithm: \( \log_a 26 - \log_a 2 = \log_a (\frac{26}{2}) = \log_a 13 \).

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Key Concepts

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

Logarithm Properties
Logarithms have unique properties that make working with them easier. One of the most important properties is the subtraction rule. This rule states that when you subtract one logarithm from another with the same base, you can combine them into a single logarithm by dividing their arguments. Specifically, \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right)\). This property is very useful for simplifying expressions.
Logarithmic Subtraction
Subtraction in logarithms generally means you are dividing the numbers inside the log. For example, in the expression \( \log_a 26 - \log_a 2\), instead of handling them individually, you leverage the subtraction property. You rearrange the expression as one single logarithm: \( \log_a \left( \frac{26}{2} \right)\). This step significantly simplifies the task.
Simplifying Logarithms
Once you've expressed the subtraction of logarithms as a single logarithm, the next task is simplification. Here, \( \log_a \left( \frac{26}{2} \right) = \log_a 13\). This is simpler and more elegant. Always look for opportunities to use properties of logarithms such as product, quotient, and power rules to make expressions more concise. Remember, \( \log_a \left( b \times c \right) = \log_a b + \log_a c\) and \( \log_a b^c = c \cdot \log_a b\) as other useful simplifications.

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