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Chapter 1: Problem 31

(a) How is the logarithmic function \(y=\log _{a} x\) defined? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the function \(\quad y=\log _{a} x\) if \(a>1\)

Short Answer

Expert verified
(a) It's the inverse of the exponential function. (b) Domain: \(x > 0\). (c) Range: all real numbers. (d) Increasing curve for \(a > 1\), passing through (1,0).

Step by step solution

01

Understanding Logarithmic Definition

The logarithmic function \(y = \log_a x\) is defined as the inverse of the exponential function \(a^y = x\). It means that for a given base \(a\) and a positive number \(x\), \(y\) is the exponent to which \(a\) must be raised to produce \(x\).
02

Determining the Domain

The domain of the logarithmic function \(y = \log_a x\) is the set of all positive real numbers \(x > 0\). This is because you cannot take the logarithm of zero or a negative number in the real number system.
03

Identifying the Range

The range of the logarithmic function \(y = \log_a x\) is the set of all real numbers. This is because \(y\) can become any real number as \(x\) varies from positive numbers near zero to extremely large values.
04

Sketching the Graph

For the function \(y = \log_a x\) with \(a > 1\), the graph is shown as a curve that passes through the point (1,0), since \(\log_a 1 = 0\). The curve is continuous, increasing, and approaches negative infinity as \(x\) approaches zero from the right. It also becomes steeper as \(x\) increases.

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Key Concepts

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

Domain of a Function
When dealing with logarithmic functions, understanding the domain is crucial. The function \(y = \log_a x\) is defined only for positive values of \(x\). This restriction exists because the logarithm of zero or any negative number is undefined in the real number system. To put it simply, for a logarithmic function to "exist," its input must always be greater than zero.

Here's why:
  • If \(x\) is zero or negative, the corresponding exponential equation \(a^y = x\) has no real solutions.
  • Logarithms reverse the exponential process, and exponentiation with positive numbers never results in non-positive values.
Therefore, the domain of \(y = \log_a x\) is \(x > 0\), which means the input must always be a positive real number.
Inverse Functions
Logarithmic functions and exponential functions are inverse operations. In simpler terms, this means they "undo" each other. When given \(y = \log_a x\), it can be rewritten as \(a^y = x\). This conversion is the core of understanding inverse functions.

A few key points to grasp:
  • For each logarithmic function, there is an equivalent exponential function, and vice versa.
  • When you apply a logarithm to an exponential function, you return to the original number. Mathematically, \( \log_a (a^y) = y \).
  • Similarly, applying an exponential function to a logarithm takes you back to the input: \( a^{\log_a x} = x \).
Knowing these principles helps to solve equations involving either logarithms or exponents by moving back and forth between forms. Mastery of inverse functions is essential in calculus and beyond.
Graph Sketching
Sketching the graph of a logarithmic function such as \(y = \log_a x\), where \(a > 1\), can be insightful into visually understanding its behavior.

Key characteristics of the graph:
  • The graph is a continuous curve.
  • It passes through the point \((1, 0)\) because for any base \(a\), the logarithm of 1 is always zero, \(\log_a 1 = 0\).
  • As \(x\) increases, \(y = \log_a x\) increases slowly. The graph gets steeper, reflecting that logarithmic growth is slow.
  • Approaching \(x = 0\) from the right, the curve plunges dramatically, heading towards negative infinity, since \(\log_a x\) is undefined for \(x \leq 0\).
This graphical representation helps in better understanding the logarithm's behavior, and it's essential for calculus, where the understanding of growth rates and limits is crucial.

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Most popular questions from this chapter

\(\begin{array}{l}{\text { If } f(x)=x+4 \text { and } h(x)=4 x-1, \text { find a function } g \text { such that }} \\ {g \circ f=h .}\end{array}\)

Graph the function \(y = x ^ { n } 2 ^ { - x } , x \geq 0 ,\) for \(n = 1,2,3,4,5\) and \(6 .\) How does the graph change as n increases?

Suppose \(f\) is even and \(g\) is odd. What can you say about \(f g ?\)

The Heaviside function \(\mathrm{H}\) is defined by $$H(t)=\left\\{\begin{array}{ll}{0} & {\text { if } t<0} \\ {1} & {\text { if } t \geqslant 0}\end{array}\right.$$ \(\begin{array}{l}{\text { It is used in the study of electric circuits to represent the sudden }} \\ {\text { surge of electric current, or voltage, when a switch is instantane- }} \\ {\text { ously turned on. }}\end{array}\) \(\begin{array}{l}{\text { (a) Sketch the graph of the Heaviside function. }} \\\ {\text { (b) Sketch the graph of the voltage V(t) in a circuit if the }} \\\ {\text { switch is turned on at time } t=0 \text { and } 120 \text { volts are applied }} \\ {\text { instantaneously to the circuit. Write a formula for } V(t) \text { in }} \\ {\text { terms of H(t). }}\end{array}\). \(\begin{array}{l}{\text { (c) Sketch the graph of the voltage } V(t) \text { in a circuit if the switch }} \\ {\text { is turned on at time } t=5 \text { seconds and } 240 \text { volts are applied }} \\ {\text { instantaneously to the circuit. Write a formula for V(t) in }} \\ {\text { terms of H(t). (Note that starting at } t=5 \text { corresponds to a }} \\ {\text { translation.) }}\end{array}\)

\(31-36\) Find the functions (a) \(\mathrm{f} \circ g,(\mathrm{b}) g \circ \mathrm{f},(\mathrm{c}) \mathrm{f} \circ \mathrm{f},\) and \((\mathrm{d}) g \circ g\) and their domains. $$f(x)=x^{2}-1, g(x)=2 x+1$$
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