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Chapter 3: Problem 27

Use the properties of logarithms to simplify the expression. $$\log _{\pi} \pi$$

Short Answer

Expert verified
The simplified form of the expression \( \log_\pi \pi \) is: 1.

Step by step solution

01

Identify the base and argument of the logarithm

Here, the base of the logarithm is \( \pi \) and the argument of the logarithm is also \( \pi \).
02

Apply the properties of logarithms

The properties of logarithms state that \( \log_b b = 1 \). So, \( \log_\pi \pi = 1 \).

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Key Concepts

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

Understanding Logarithms
Logarithms are incredibly useful in mathematics for expressing how one number relates to another through exponentiation. A logarithm of a certain base is the power to which the base must be raised to achieve a particular number. For example, if you have an expression \( \log_b a \), it is asking: "To what power must \( b \) be raised to give \( a \)?". The result is the exponent.
  • The expression \( \log_2 8 \) equals 3, because \( 2^3 = 8 \).
  • The expression \( \log_{10} 100 \) equals 2, because \( 10^2 = 100 \).
Logarithms make complex calculations simpler, particularly when dealing with exponential growth or decay. They also appear in many scientific fields, including physics and finance.
Base and Argument in Logarithms
In any logarithmic expression, such as \( \log_b a \), there are two main components: the base and the argument.
  • Base (\( b \)): The base is the number we are raising to a power. In many cases, like logarithms used in specific scientific or engineering contexts, the base is consistent, such as 10 in common logarithms or \( e \) in natural logarithms.
  • Argument (\( a \)): The argument is the number for which we want to find the logarithm. It is the result of raising the base to a certain power.
Understanding the base and argument is crucial because the logarithmic properties and laws often depend on these components remaining consistent. It helps determine the value of the logarithm, such as knowing why \( \log_\pi \pi \) simplifies to 1.
Simplification of Logarithmic Expressions
The simplification of logarithmic expressions involves using logarithmic properties to make an expression easier to understand or evaluate. One of the most direct properties we use frequently is the rule that states \( \log_b b = 1 \). This tells us that if the base and the argument of a logarithm are the same, the log value is simply 1.
  • For example, \( \log_5 5 \) equals 1, because 5 raised to the power of 1 is 5.
  • Similarly, \( \log_8 8 \) equals 1, because 8 raised to the power of 1 is 8, just like any other matching base and argument.
Knowing this property is key to quickly simplify expressions without detailed calculations. It's one of the fundamental tools for tackling logarithmic expressions efficiently.

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Most popular questions from this chapter

Determine whether the statement is true or false given that \(f(x)=\ln x .\) Justify your answer. $$f(x-2)=f(x)-f(2), \quad x>2$$

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Use the following information. The relationship between the number of decibels \(\boldsymbol{\beta}\) and the intensity of a sound \(I\) in watts per square meter is given by $$\boldsymbol{\beta}=10 \log \left(\frac{\boldsymbol{I}}{\mathbf{1 0}^{-12}}\right).$$ Find the difference in loudness between an average office with an intensity of \(1.26 \times 10^{-7}\) watt per square meter and a broadcast studio with an intensity of \(3.16 \times 10^{-10}\) watt per square meter.

Find a logarithmic equation that relates \(y\) and \(x .\) Explain the steps used to find the equation. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline y & 2.5 & 2.102 & 1.9 & 1.768 & 1.672 & 1.597 \\ \hline\end{array}$$
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