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Chapter 4: Problem 3

Simplify the following expressions. $$\frac{1}{2} \ln 9$$

Short Answer

Expert verified
The simplified expression is \[ \text{ln}(3) \]

Step by step solution

01

Apply the Property to the Expression

We can rewrite \[ \frac{1}{2} \times \text{ln}(9) = \text{ln}(9^{\frac{1}{2}}) \]
02

Simplify the Expression

Since \[ 9^{\frac{1}{2}} = \text{the square root of}\text{ln}(3^{2}) 3^{2 *\frac{1}{2}}= 3 \]we can replace the 9 with 3:
03

Write the Final Answer

Therefore, \[ \text{ln}(9^{\frac{1}{2}}) = \text{ln}(3) \]

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Key Concepts

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

Properties of Logarithms
Understanding logarithmic simplification starts with knowing the properties of logarithms. These properties help in breaking down complex logarithmic expressions into simpler forms.
Some key properties include:
  • Product Property: \(\text{ln}(ab) = \text{ln}(a) + \text{ln}(b)\)
  • Quotient Property: \(\text{ln}(\frac{a}{b}) = \text{ln}(a) - \text{ln}(b)\)
  • Power Property: \(\text{ln}(a^b) = b \times \text{ln}(a)\)
In the given exercise, the Power Property is used. We start with \( \frac{1}{2} \times \text{ln}(9)\) and transform it to \[ \text{ln}(9^{\frac{1}{2}}) \]. This uses the property which states logarithm of a number raised to a power can be expressed as the power times the logarithm of the number.
Natural Logarithm
The natural logarithm, denoted as \( \text{ln} \), is a logarithm to the base \( e \). \( e \) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm has some unique properties which make it very useful in calculus and real-world applications.
  • If \( \text{ln}(e) = 1 \)
  • \( \text{ln}(1) = 0 \)
In the exercise provided, \( \text{ln}(9) \) is a part of the simplification process. An understanding of natural logarithms is crucial for simplifying and evaluating expressions involving \( \text{ln} \). When we write \( \text{ln}(9^{1/2}) \), we are essentially finding the natural logarithm of 9 raised to the power of 1/2, further breaking down the calculation.
Exponentiation
Exponentiation is a mathematical operation that involves raising a number (the base) to the power of an exponent. For example, \( 9^{1/2} \) represents the square root of 9, because any number to the power of 1/2 is its square root. This is essential in our exercise where we have \( 9^{1/2} = 3 \).
Here's a breakdown:
  • \( 9 = 3^2 \)
  • \( 9^{1/2} = (3^2)^{1/2} \)
  • \( (3^2)^{1/2} = 3 \)
This simplification helps in transforming \( \text{ln}(9^{1/2}) \) to \( \text{ln}(3) \). Understanding exponentiation aids in simplifying various logarithmic expressions quickly and effectively, as shown in the provided solution.

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