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Chapter 4: Problem 19

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \sqrt[5]{x} $$

Short Answer

Expert verified
The expanded form of \( \ln \sqrt[5]{x} \) is \( \frac{1}{5}\ln x \).

Step by step solution

01

Identify the property of logarithm

Look at the logarithmic expression and identify the properties of logarithms applicable here. The expression is a natural logarithm of a fifth root of some number x. The specific logarithmic property we will use here is that the logarithm of a root is the same as dividing the logarithm of the number by the root. Therefore, \( \ln \sqrt[5]{x} = \frac{1}{5}\ln x \).
02

Apply the Logarithmic Property

Next step is applying the logarithmic property on \( \ln \sqrt[5]{x} \). Rewrite \( \sqrt[5]{x} \) as \( x^{1/5} \). Now applying the logarithmic property, coefficient of \( x^{1/5} \) becomes the multiplier to \( \ln x \). So, the expanded form of \( \ln \sqrt[5]{x} = \frac{1}{5}\ln x \).

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Most popular questions from this chapter

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 3 \ln x+5 \ln y-6 \ln z $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log 5+\log 2 $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log (1000 x) $$

Without showing the details, explain how to condense \(\ln x-2 \ln (x+1)\)

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{2}(\log x+\log y) $$
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