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Magazine Chapter 4: Problem 3
Find each logarithm without using a calculator or tables. a. \(\ln \left(e^{10}\right)\) b. \(\ln \sqrt{e}\) c. \(\ln \sqrt[3]{e^{4}}\) d. \(\ln 1\) e. \(\ln \left(\ln \left(e^{e}\right)\right)\) f. \(\ln \left(\frac{1}{e^{3}}\right)\)
Short Answer
Expert verified
a. 10, b. 1/2, c. 4/3, d. 0, e. 1, f. -3.
Step by step solution
01
Simplify the Argument for Part a
Given \( \ln \left(e^{10}\right) \). Use the property of logarithms that says \( \ln(e^{x}) = x \). Substitute \( x = 10 \) to get \( \ln(e^{10}) = 10 \).
02
Simplify the Argument for Part b
Given \( \ln \sqrt{e} \). Rewrite \( \sqrt{e} \) as \( e^{1/2} \). Then, apply the property \( \ln(e^{x}) = x \) to get \( \ln(e^{1/2}) = 1/2 \).
03
Simplify the Argument for Part c
Given \( \ln \sqrt[3]{e^{4}} \). Rewrite \( \sqrt[3]{e^{4}} \) as \( e^{4/3} \). Using the property \( \ln(e^{x}) = x \), \( \ln(e^{4/3}) = 4/3 \).
04
Logarithm of One for Part d
Given \( \ln 1 \), use the property that the natural logarithm of 1 is always 0. Therefore, \( \ln 1 = 0 \).
05
Simplify Logarithm within Logarithm for Part e
Given \( \ln \left(\ln \left(e^{e}\right)\right) \). First, solve \( \ln(e^{e}) \) using the property \( \ln(e^{x}) = x \), thus \( \ln(e^{e}) = e \). Now, \( \ln(e) = 1 \). Therefore, \( \ln(\ln(e^{e})) = \ln(e) = 1 \).
06
Simplify Negative Exponent for Part f
Given \( \ln \left(\frac{1}{e^{3}}\right) \). Rewrite as \( \ln(e^{-3}) \). Using \( \ln(e^{x}) = x \), we have \( \ln(e^{-3}) = -3 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Natural Logarithms
The natural logarithm, denoted as \( \ln \), is a special type of logarithm where the base is the mathematical constant \( e \). The number \( e \) is approximately 2.71828, and it's an important constant in mathematics because it arises naturally in various contexts, especially in calculus and exponential growth.
Natural logarithms are specifically used to simplify expressions involving exponential growth and decay, and they occur frequently in science and engineering problems.
Here's a summary to make things easier:
Natural logarithms are specifically used to simplify expressions involving exponential growth and decay, and they occur frequently in science and engineering problems.
Here's a summary to make things easier:
- Natural logarithms find out how many times \( e \) must be multiplied by itself to reach a certain number.
- The property \( \ln(e^{x}) = x \) allows us to simplify expressions when working with exponents that have a base of \( e \).
Exploring Logarithmic Properties
Logarithms have several properties that make them powerful tools for mathematical simplification. These properties help in solving equations and transforming expressions. Here are some fundamental logarithmic properties:
- The identity property: For any base, the log of \( 1 \) is always \( 0 \) because any number to the power of \( 0 \) equals \( 1 \). Hence, \( \ln(1) = 0 \).
- Power rule: \( \ln(a^b) = b \ln(a) \), allowing us to pull out any exponent as a multiplicative factor.
- Multiplicative property: \( \ln(ab) = \ln(a) + \ln(b) \).
- Division property: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
The Role of Exponents in Logarithms
Exponents indicate how many times a number, known as the base, is multiplied by itself. They are the key focus when it comes to simplifying logarithmic expressions, especially with the natural logarithm.
The natural logarithm has a special property related to exponents: if you have an exponent in the expression, like \( e^{x} \), the natural logarithm \( \ln(e^{x}) \) will directly simplify to \( x \). This is immensely useful when working to simplify expressions, as seen in the steps for parts a, b, and c.
For example:
The natural logarithm has a special property related to exponents: if you have an exponent in the expression, like \( e^{x} \), the natural logarithm \( \ln(e^{x}) \) will directly simplify to \( x \). This is immensely useful when working to simplify expressions, as seen in the steps for parts a, b, and c.
For example:
- Simplification from powers: By recognizing powers of \( e \), such as \( \ln(e^{10}) \), it simplifies straight to \( 10 \).
- Fractional exponents: When managing square and cube roots like \( \sqrt{e} \) and \( \sqrt[3]{e^{4}} \), rewriting them as powers \( e^{1/2} \) or \( e^{4/3} \) makes the solution straightforward using logarithmic properties.
Demystifying Mathematical Simplification
Mathematical simplification is a process of transforming expressions into their most fundamental or manageable form, making it easier to work with and understand.
When dealing with logarithmic expressions, such as those involving natural logarithms and exponents, simplification often involves applying the logarithmic properties and exponent rules we discussed earlier.
The steps often include:
When dealing with logarithmic expressions, such as those involving natural logarithms and exponents, simplification often involves applying the logarithmic properties and exponent rules we discussed earlier.
The steps often include:
- Using identities: Quickly simplify questions like \( \ln(1) = 0 \) due to identity properties.
- Breaking down expressions: Such as taking complex statements like \( \ln \left( \frac{1}{e^3} \right) \) and rewriting it as \( \ln(e^{-3}) \), which simplifies directly to \( -3 \).
- Layer removal: In scenarios like \( \ln\left(\ln\left(e^{e}\right)\right) \), where using \( \ln(e^{e}) = e \) reduces in complexity to finally just \( 1 \).
