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Chapter 3: Problem 46

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{3} 405-\log _{3} 5$$

Short Answer

Expert verified
The transformed expression is \(\log_{3}{81} = 4\)

Step by step solution

01

Analyzing the Problem

Firstly, review the properties of logarithms. For this task, the critical property is that the subtraction of two logs with the same base is equivalent to the log of the division between the original values.
02

Applying the Logarithm Division Rule

Utilize the division rule to simplify the expression. Given the formula \(\log_a{m} - \log_a{n} = \log_a{\frac{m}{n}}\), replace \(m\) with 405 and \(n\) with 5: \(\log _{3} 405 - \log _{3} 5 = \log_{3}{\frac{405}{5}}\)
03

Evaluating the Division

Divide 405 by 5 to get 81: \(\log_{3}{\frac{405}{5}} = \log_{3}{81}\)
04

Evaluating Logarithm

The question states that evaluation should be done without a calculator where possible. Since \(3^4 = 81\), it can be stated that \(\log_{3}{81} = 4\) . Hence, the equation \(\log_{3}{81} = 4\) is true.

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Most popular questions from this chapter

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=4.5(0.6)^{x}$$

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