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Chapter 3: Problem 2

Fill in the blanks. The common logarithmic function has base __________.

Short Answer

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10

Step by step solution

01

Identify the Characteristics of the Common Logarithmic Function

The common logarithmic function, typically written as \(log(x)\), is a specific type of logarithmic function that has a fixed base.
02

Recall the Base of the Common Logarithmic Function

In mathematics, the common logarithmic function uses the constant \(10\) as its base. This is a standard convention in logarithms and is therefore important to remember.

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Key Concepts

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

Logarithmic Function
A logarithmic function is a mathematical function used to determine the exponent or power to which a base number must be raised to obtain another number. In simpler terms, if you have an equation like \( b^y = x \), the logarithm base \( b \) of \( x \) would be \( y \). This is written as \( \log_b(x) = y \). Logarithmic functions are the inverses of exponential functions, meaning they undo the operations of exponentials.
  • A key property of logarithms is that they convert multiplication into addition. For example, \( \log_b(mn) = \log_b(m) + \log_b(n) \).
  • Logarithms offer a way to address problems involving multiplicative relationships and exponential growth more easily.
They are used extensively in various fields such as science, engineering, and finance. From computing compound interest to measuring the intensity of earthquakes in seismology, logarithmic functions serve a broad range of practical applications.
Base 10
A base 10 logarithm, also known as the common logarithm, is a logarithmic function where the base is set to 10. This is typically written as \( \log(x) \), with the base implied rather than explicitly stated. Using base 10 is a standard because of its ease of use in calculations and its historical connection to our decimal numeric system.
  • It simplifies tasks that deal with orders of magnitude, such as scales used in measuring sound intensity (decibels).
  • Common logarithms make it easier to quickly compare very large or very small numbers by reducing these large ranges to more manageable figures.
Base 10 is a natural choice for logarithms in many applications, especially in cases where human-readable and comprehensible scales are needed.
Mathematics
In mathematics, understanding logarithmic functions is crucial because they provide practical solutions to many real-world problems, especially those involving exponential growth or decay. Problems such as calculating interest, population growth, light intensity, and pH levels in chemistry are often framed in terms of logs.
  • The notation \( \log_b(x) \) means you're asking: "To what power must the base \( b \) be raised to produce \( x \)?"
  • For equations where the base is unstated, 10 is assumed, making it essential to grasp the concept of the common logarithm.
Grasping the concept of logarithms allows mathematical ideas to be expressed in a form that is much simpler to analyze, making them a necessary tool for both students and professionals in numerous fields. This core concept extends beyond general mathematics; it enhances logical thinking and problem-solving skills.

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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. If \(f(x)<0,\) then \(0

Rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. The logarithm of the product of two numbers is equal to me sum of the logarithms of the numbers.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{4} x-\log _{4}(x-1)=\frac{1}{2}$$

Compare the logarithmic quantities. If two are equal, then explain why. $$\log _{7} \sqrt{70}, \quad \log _{7} 35, \quad \frac{1}{2}+\log _{7} \sqrt{10}$$

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$3-\ln x=0$$
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