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Chapter 1: Problem 17

For \(b>0\) with \(b \neq 1,\) what are the domain and range of \(f(x)=\log _{b} x\) and why?

Short Answer

Expert verified
Question: Determine the domain and range of the logarithmic function \(f(x) = \log_b x\), where b > 0 and b 鈮� 1. Answer: The domain of the function \(f(x) = \log_b x\) is (0, +鈭�), and the range is (-鈭�, +鈭�).

Step by step solution

01

Analyze the properties of logarithms.

Logarithmic functions have a base greater than 0, but the base cannot be equal to 1. Therefore, \(b>0\) and \(b\neq1\). In addition, logarithms are not defined for input values less than or equal to 0, as the logarithm function represents the power to which the base must be raised to obtain the input.
02

Determine the domain of the function.

Given the properties of logarithms, we need \(x > 0\) for the function \(f(x) = \log_b x\) to be defined. So, the domain of the function is the set of all positive real numbers, which can be written in interval notation as \((0,+\infty)\).
03

Determine the range of the function.

Recall that the logarithmic function \(f(x) = \log_b x\) is the inverse of the exponential function \(g(x) = b^x\). The exponential function has a range of \((0,+\infty)\) and is strictly increasing (or decreasing) when \(b>1\) (or \(0<b<1\)), so its inverse function, the logarithm, will have a domain that is the range of the exponential function. Thus, the range of the logarithmic function will be the set of all real numbers, or \((-\infty, +\infty)\).
04

Write the final answer.

The domain of the function \(f(x) = \log_b x\) is \((0,+\infty)\), and the range is \((-\infty, +\infty)\).

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Key Concepts

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

Domain of a Function
Understanding the domain of a function is crucial in mathematics as it tells us the set of input values for which the function is defined. For logarithmic functions, such as \(f(x) = \log_b x\), where \(b>0\) and \(b eq 1\), the domain specifically relates to the permissible x-values that keep the function valid. Logarithms are only defined for positive numbers. This is because a logarithm, like \(\log_b x\), asks the question: "To what power should we raise the base \(b\) to get \(x\)?"

  • Since raising any positive number \(b\) to a real power can't result in zero or a negative number, the input \(x\) must be greater than 0.
  • Therefore, the domain of the function \(f(x) = \log_b x\) is all positive real numbers.
In interval notation, we express this domain as \((0, +\infty)\). This indicates that the function works for all numbers greater than zero but not including zero itself.
Range of a Function
The range of a function describes the set of possible output values the function can produce. For the logarithmic function \(f(x) = \log_b x\), the outputs are all real numbers. This is due to its relationship with its inverse, the exponential function.

  • An exponential function \(g(x) = b^x\) can generate any positive real number as its output.
  • Since the logarithmic function is the inverse of the exponential function, it can correspondingly yield any real number as its result.
The range of the logarithmic function \(f(x) = \log_b x\) includes all real numbers, written in interval notation as \((-\infty, +\infty)\). So, any real value is possible from this function, which makes the logarithmic output set quite expansive.
Inverse Functions
Inverse functions provide a fascinating way to "reverse" a given operation. When we talk about an inverse function, we refer to a function that "undoes" the effect of the original function. For logarithmic functions like \(f(x) = \log_b x\), the inverse is the exponential function \(g(x) = b^x\). This is a key point:

  • If you apply a logarithm to a number, and then exponentiate it, you'll get back to the original number, assuming the base is the same.
  • This connection between logarithms and exponentials means understanding one can help you understand the other.
To illustrate further, think about it this way: if \(y = \log_b x\), then \(b^y = x\). So, if you know the logarithm, you can determine the equivalent power in the exponential form. Inverses like these are incredibly useful, not just for solving equations but for uncovering the relationships between different mathematical functions.

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Most popular questions from this chapter

Show that the graph of \(f(x)=10+\sqrt{-x^{2}+10 x-9}\) is the upper half of a circle. Then determine the domain and range of the function.

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