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Chapter 5: Problem 15

Express as a product. $$\ln \sqrt[3]{4}$$

Short Answer

Expert verified
\( \ln \sqrt[3]{4} = \frac{1}{3} \ln(4) \)

Step by step solution

01

- Understand the Problem

The problem requires expressing \( \ln \sqrt[3]{4} \) as a product.
02

- Use Properties of Logarithms

Recall that the natural logarithm of a number with an exponent can be rewritten using the property \( \ln(a^b) = b \ln(a) \).
03

- Rewrite the Expression

Identify that \( \sqrt[3]{4} \) is the same as \( 4^{1/3} \). Using the logarithm property, we get \( \ln(4^{1/3}) = \frac{1}{3} \ln(4) \).
04

- Express as a Product

The expression \( \frac{1}{3} \ln(4) \) can be written as a product of \( \frac{1}{3} \) and \( \ln(4) \). Thus, \( \ln \sqrt[3]{4} = \frac{1}{3} \ln(4) \).

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Key Concepts

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

Natural Logarithm
The term **natural logarithm** is often abbreviated as \(\text{ln}\). It's the logarithm to the base **e**, where **e** is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm of a number **x** is the power to which **e** must be raised to get **x**. For example, \(\text{ln}(e) = 1\) because \(\text{e}^1 = e\). Similarly, \(\text{ln}(1) = 0\) because \(\text{e}^0 = 1\).

In this exercise, we started with \(\text{ln} \sqrt[3]{4}\). The natural logarithm is key in simplifying expressions involving exponents and roots.

Remember, when you see \(\text{ln}\), think about the relationship between the number and **e**.
Logarithmic Properties
Logarithmic properties are essential tools for manipulating and simplifying logarithmic expressions. One important property used in this problem is:

\( \text{ln}(a^b) = b \text{ln}(a) \)

This property allows you to take the exponent in the argument of a natural logarithm and move it to the front as a multiplier. It simplifies expressions drastically. For example, \(\text{ln}(4^{1/3}) = \frac{1}{3} \text{ln}(4)\).

Other important logarithmic properties include:
  • \(\text{ln}(ab) = \text{ln}(a) + \text{ln}(b)\)
  • \(\text{ln}(a/b) = \text{ln}(a) - \text{ln}(b)\)
These properties help in breaking down complex logarithmic expressions into simpler, more manageable parts. Understanding and mastering these properties is crucial in algebra and calculus.
Exponents
An exponent tells you how many times to multiply a number by itself. In this problem, we dealt with a cube root, which is also an exponent. The cube root of 4, written as \(\text{\text{鈭殅}[3]{4}\), is the same as \(\text{4}^{1/3}\).

Key points to remember about exponents:
  • \((a^m)^n = a^{mn}\)
  • \(\text{鈭殅[n]{a} = a^{1/n}\)
  • \((a^m \times a^n) = a^{m+n}\)
By using these rules, we simplified \(\text{鈭殅[3]{4}\) into the form \(\text{4}^{1/3}\). This form helps us apply the logarithmic property \(\text{ln}(a^b) = b \text{ln}(a)\), making it easier to express as a product. The final step made use of this concept efficiently.

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Most popular questions from this chapter

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\log _{a} x^{3}=3 \log _{a} x$$

Express as a single logarithm and, if possible, simplify. $$\log _{a}\left(x^{2}+x y+y^{2}\right)+\log _{a}(x-y)$$

Solve using any method. $$\left(\log _{3} x\right)^{2}-\log _{3} x^{2}=3$$

Express as a sum or a difference of logarithms. $$\log _{a} \sqrt{9-x^{2}}$$

Centenarian Population. The centenarian population in the United States has grown over \(65 \%\) in the last 30 years. In \(1980,\) there were only \(32,194\) residents ages 100 and over. This number had grown to \(53,364\) by \(2010 .\) (Sources: Population Projections Program; U.S. Census Bureau; U.S. Department of Commerce; "What People Who Live to 100 Have in Common," by Emily Brandon, U.S. News and World Report, January \(7,2013\) ) The exponential function $$ H(t)=80,040.68(1.0481)^{t} $$ where \(t\) is the number of years after \(2015,\) can be used to project the number of centenarians. Use this function to project the centenarian population in 2020 and in 2050 (IMAGE CANT COPY)
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