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Chapter 9: Problem 8

Prove the given property if \(a\) is any positive number and \(x\) and \(y\) are any positive numbers. $$ \log _{a} 1=0 $$

Short Answer

Expert verified
\(\text{log}_a 1 = 0\) because \(a^0 = 1\).

Step by step solution

01

Understand the logarithmic property

The property to prove is \(\text{log}_a 1 = 0\). This means that the logarithm of 1 to any positive base \(a\) is 0.
02

Recall the definition of logarithms

By definition, \(\text{log}_a b = c\) implies \(a^c = b\). Here, \(a > 0\) and \(a eq 1\).
03

Apply the definition to the given property

For \(\text{log}_a 1\), set \(b = 1\). We need to find the value of \(c\) such that \(a^c = 1\).
04

Solve for the exponent

To satisfy \(a^c = 1\) for any positive base \(a\), the exponent \(c\) must be 0, because any number raised to the power of 0 is 1. Therefore, \(c = 0\).
05

Conclude the proof

Since \(\text{log}_a 1 = 0\) meets the condition that \(a^0 = 1\), we have proven the property.

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Key Concepts

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

logarithm definition
A logarithm is a mathematical operation that is the inverse of exponentiation. If you have an equation in the form of \(a^c = b\), the logarithm helps you find the exponent \(c\). The definition of a logarithm can be written as: \(\log_a b = c\), which means that \(a^c = b\).
Here is how it works:
  • For a base \(a\), which is a positive number and not equal to 1.
  • \(b\), another positive number.
  • \(\log_a b = c\) helps you determine the power \(c\) to which you need to raise \(a\) to get \(b\).
The logarithm essentially answers the question: 'To what power must the base \(a\) be raised to produce the number \(b\)?'
In simpler terms, it's a tool for solving equations where the unknown variable is an exponent.
logarithmic function
A logarithmic function is simply a function that uses logarithms. The general form of a logarithmic function is \(f(x) = \log_a(x)\), where \(a\) is the base of the logarithm. Logarithmic functions are useful for modeling many natural phenomena, including exponential growth and decay.
Important points to note about logarithmic functions include:
  • They are the inverse of exponential functions.
  • The domain of a logarithmic function is positive real numbers (\(x > 0\)).
  • The range of a logarithmic function is all real numbers.
  • They pass through the point (1, 0) because \(\log_a 1 = 0\), as shown in our original exercise.

These functions have several important properties:
  • \(\log_a (xy) = \log_a x + \log_a y\)
  • \(\log_a \left( \frac{x}{y} \right) = \log_a x - \log_a y\)
  • \(\log_a (x^k) = k \log_a x\)

Understanding these properties makes it easier to manipulate and solve logarithmic equations.
exponentiation
Exponentiation is a mathematical operation that involves raising a number to the power of another number. If you have a base \(a\) and an exponent \(c\), exponentiation is written as \(a^c\), which means \(a\) is multiplied by itself \(c\) times.
Here are key points about exponentiation:
  • It serves as the basis for defining logarithms.
  • Any positive number raised to the power of 0 is 1 (\(a^0 = 1\)).
  • Exponentiation with a negative exponent represents a reciprocal (\(a^{-c} = \frac{1}{a^c}\)).

Exponentiation follows several important rules:
  • \(a^m \cdot a^n = a^{m+n}\)
  • \(\frac{a^m}{a^n} = a^{m-n}\)
  • \((a^m)^n = a^{mn}\)

Understanding these rules is crucial for manipulating exponential expressions and solving equations. Just like in the original exercise, where we saw that any number raised to 0 equals 1, this is a fundamental property of exponents.

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Most popular questions from this chapter

Prove that \(x-1-\ln x>0\) and \(1-\ln x-1 / x<0\) for all \(x>0\) and \(x \neq 1\), thus establishing the inequality $$ 1-\frac{1}{x}<\ln x0\) and \(x \neq 1\). (HINT: Let \(f(x)=x-1-\ln x\) and \(g(x)=1-\ln x-1 / x\) and determine the signs of \(f^{\prime}(x)\) and \(g^{\prime}(x)\) on the intervals \((0,1)\) and \(\left.(1,+\infty) .\right)\)

If \(f(x)=1 / x\) find the average value of \(f\) on the interval \([1,5]\)

\(\int \frac{d x}{x \ln x}\)

\(\int_{1}^{2} \frac{e^{x}}{e^{x}+e} d x\)

\(y=e^{-3 x^{2}}\)
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