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Chapter 3: Problem 89

Writing Explain why \(\log _{a} x\) is defined only for \(01\) Why vasylehutterstock com

Short Answer

Expert verified
The logarithm function \(\log _{a} x\) is defined only for \(01\) because in these cases, the corresponding exponential function \(a^x\) is a one-to-one and onto function that is defined for all real \(x\), which allows it to have an inverse function. Meanwhile, for \(a=1\) or \(a鈮�0\), the corresponding exponential function is either not one-to-one or not defined for all real numbers, hence does not have an inverse, making the logarithm base \(a\) undefined.

Step by step solution

01

Explanation of Logarithm function

Firstly, it is important to note that the logarithmic function \(\log _{a} x\) is the inverse of the exponential function \(a^x\). Meaning, if \(y=a^x\), then \(x=\log _{a} y\).
02

Discussing the Case of \(a=1\)

If \(a=1\), the exponential function \(1^x\) would be a constant function, not a one-to-one or onto function. Therefore, it doesn't have an inverse, and consequently, \(\log _{1} x\) is undefined.
03

Discussing the Case of \(a鈮�0\)

If \(a鈮�0\), the exponential function is not well-defined for all real numbers, nor is it one-to-one function, so it also would not have an inverse. Hence, the logarithm base \(a鈮�0\) is undefined.
04

Discussing the Case of \(0

Conversely, if \(01\), the exponential function \(a^x\) is a one-to-one and onto function. It covers all positive value of \(y\) for all \(x\). Hence, its inverse, the logarithm function \(\log _{a} x\), is also defined for these ranges of \(a\). This demonstrates why \(\log _{a} x\) is defined only for \(01\).

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Key Concepts

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

Understanding the Exponential Function
The exponential function is a mathematical concept where a constant base is raised to the power of a variable exponent. In the form of \( a^x \), the base \( a \) is a positive real number. The variable, represented as \( x \), can take any real value.

Key properties of exponential functions include:
  • The base of the function, \( a \), must be a positive number.
  • The function exhibits rapid growth if \( a > 1 \) and gradual decay if \( 0 < a < 1 \).
  • It is always positive regardless of the value of \( x \).
  • These functions are commonly used to model growth processes, such as population growth or compound interest.
A properly defined exponential function is both one-to-one and onto when it covers a specific range of outputs for given bases. Understanding these properties helps in determining how and why their inverse functions, the logarithms, are defined.
The Inverse Function Concept
An inverse function is a function that "reverses" the effect of the original function. For a function \( f(x) \), its inverse is denoted as \( f^{-1}(x) \). The main idea is that if \( f(x) = y \), then \( f^{-1}(y) = x \).

For logarithmic functions, the inverse of the exponential function \( a^x \) is the logarithmic function \( \log_a(x) \). It means if \( y = a^x \), then \( x = \log_a(y) \).

Key points to understand an inverse function:
  • To find an inverse function, swap the roles of \( x \) and \( y \) in the equation.
  • An exponential function is invertible if it is a one-to-one function.
  • Logarithms provide a mechanism to find the power needed to obtain a certain number from a given base.
By understanding the relationship between an exponential function and its inverse, one can effectively solve equations involving these mathematical concepts.
The Nature of One-to-One Functions
A function is one-to-one if each value of the output variable corresponds uniquely to a value of the input variable. In simple terms, no two different inputs produce the same output. This implies strict monotonicity 鈥� the function is either always increasing or always decreasing.

For the exponential function \( a^x \):
  • It is one-to-one when \( a > 0 \).
  • For \( a = 1 \), the function is a constant (not one-to-one), hence can't have a logarithmic inverse.
  • In the range \( 0 < a < 1 \), the function is decreasing and still one-to-one.
One-to-one functions are crucial in defining inverses because they ensure each input maps to a distinct output. This is why the logarithm can exist only when the base \( a \) results in such a strictly monotonic function.
Understanding Onto Functions
A function is called onto, or surjective, if every possible output value is covered by the function. For an exponential function \( a^x \), the range is all real numbers greater than zero.

Exploring onto functions in the context of exponential functions:
  • The function \( a^x \) for \( a > 1 \) or \( 0 < a < 1 \) maps to all positives, making it onto when considering positive real numbers.
  • Both cases ensure all positive values of \( y \) have pre-images, which are necessary for defining its inverse, the logarithmic function.
  • If there are outputs that cannot be achieved, the function is not onto for that range.
For the logarithmic function \( \log_a(x) \), being the inverse of an onto function means it can match each positive value of \( x \) to a real number, consolidating its definition when \( 0 < a < 1 \) or \( a > 1 \).

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Most popular questions from this chapter

Writing a Natural Logarithmic Equation In Exercises \(53-56,\) write the exponential equation in logarithmic form. $$e^{1 / 2}=1.6487 \ldots$$

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$6 e^{1-x}=25$$

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 the sum of the logarithms of the numbers.

Function \(\quad\) Value \(\begin{array}{ll}\text { 57. } f(x)=\ln x & x=18.42 \\ \text { 58. } f(x)=3 \ln x & x=0.74\end{array}\) \(\begin{array}{lll}y & f(x)=3 \ln x & x=0.74 \\ \text { 59. } g(x)=8 \ln x & x=0.05\end{array}\) 60\. \(g(x)=-\ln x \quad x=\frac{1}{2}\)

Use the acidity model given by \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right],\) where acidity \((\mathbf{p H})\) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. The \(\mathrm{pH}\) of a solution decreases by one unit. By what factor does the hydrogen ion concentration increase?
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