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Chapter 5: Problem 72

Find the logarithm using common logarithms and the change-of-base formula. $$\log _{\pi} 100$$

Short Answer

Expert verified
Approximately 4.025

Step by step solution

01

- Understand the Change-of-Base Formula

To find the logarithm \(\text{log}_{\text{base}} x\) using common logarithms (logarithms with base 10), use the change-of-base formula. The formula is: \[ \text{log}_{b} x = \frac{\text{log}_{10} x}{\text{log}_{10} b} \]
02

- Apply the Formula

Replace \(b\) with \(\text{蟺}\) and \(x\) with \(100\). According to the formula: \[ \text{log}_{\text{蟺}} 100 = \frac{\text{log}_{10} 100}{\text{log}_{10} 蟺} \]
03

- Compute Common Logarithms

Now, calculate the common logarithms. \[ \text{log}_{10} 100 = 2 \] because \(\text{10}^2 = 100\). Next, use a calculator for \(\text{log}_{10} 蟺\): \[ \text{log}_{10} 蟺 \approx 0.497 \] (rounded to three decimal places)
04

- Divide the Values

Divide the common logarithms obtained: \[ \text{log}_{\text{蟺}} 100 = \frac{2}{0.497} \approx 4.025 \]

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Key Concepts

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

Common Logarithms
Common logarithms are logarithms with base 10. They are often written as \(\text{log}_{10} x\) or simply \(\text{log} x\). This notation is useful because it simplifies many calculations, especially when dealing with decimal numbers. Common logarithms appear frequently in scientific calculations and various fields such as engineering, physics, and data analysis.
For instance, if you need to find \(\text{log}_{10} 100\), you are determining the power to which 10 must be raised to get 100. This power is 2, since \(10^2 = 100\). Therefore, \(\text{log}_{10} 100 = 2\).
This simplicity makes common logarithms a foundational tool in understanding more complex logarithmic functions.
Logarithmic Functions
Logarithmic functions are the inverse operations of exponential functions. If you have an exponential function like \( y = b^x \), the logarithmic function is expressed as \( x = \text{log}_b y \). This means that \(\text{log}_b y\) finds the power that \(b\) must be raised to result in \(y\).
For example:
  • \( y = 10^2 \) can be rewritten as \(2 = \text{log}_{10} y \) which simplifies to \(\text{log}_{10} 100 = 2\).
  • The notation is crucial because it generalizes the concept of finding unknown exponents across different bases.
Logarithmic functions help in solving exponential growth and decay problems, making them vital in disciplines like biology, chemistry, and economics.
Logarithmic Properties
Understanding properties of logarithms aids in simplifying and solving logarithmic equations. Here are some key properties:
  • **Product Property**: \(\text{log}_b (xy) = \text{log}_b x + \text{log}_b y \)
  • **Quotient Property**: \(\text{log}_b \frac{x}{y} = \text{log}_b x - \text{log}_b y \)
  • **Power Property**: \(\text{log}_b (x^c) = c \text{log}_b x \)
  • **Change of Base Formula**: \(\text{log}_b x = \frac{\text{log}_k x}{\text{log}_k b}\) - this is essential for converting logarithms with one base to another, as seen in the original exercise when converting \(\text{log}_{\text{蟺}} 100 \) to common logarithms.
These properties enable you to break down complex logarithmic expressions into simpler parts, making them easier to manipulate and solve.

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