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Chapter 9: Problem 27

Express as an equivalent expression that is a single logarithm. $$\log _{a} 19-\log _{a} 2$$

Short Answer

Expert verified
\(\log_a \left(\frac{19}{2}\right)\)

Step by step solution

01

Identify the Logarithmic Expression

The given expression to simplify is \(\log_a 19 - \log_a 2\).
02

Apply the Logarithmic Property

Recall that the difference of two logs with the same base can be written as the log of a quotient. The property is: \(\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)\).
03

Utilize the Property

Apply the property to the expression: \(\log_a 19 - \log_a 2 = \log_a \left(\frac{19}{2}\right)\).
04

Write the Final Answer

The simplified expression as a single logarithm is \(\log_a \left(\frac{19}{2}\right)\).

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Key Concepts

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

logarithmic expressions
A logarithmic expression involves the logarithm of a number. A logarithm answers the question: 'To what exponent must the base be raised, to produce a given number?' We denote it as \(\text{log}_b M\), where 'b' is the base and 'M' is the number we want to find the logarithm for. For example, \(\text{log}_a 19\) asks, 'a raised to which power equals 19?'. Logarithms can use any positive number as a base, but common ones include 10 (common logarithm), 'e' (natural logarithm), and 2 (binary logarithm). Understanding the structure of a logarithmic expression helps in applying logarithmic properties.
difference of logarithms
When facing an exercise where you need to simplify a logarithmic expression such as \(\text{log}_a 19 - \text{log}_a 2\), you can directly apply this property. Here, you would write:

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