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Chapter 13: Problem 57

Perform the indicated operations. $$\text { If } f(x)=\log _{5} x, \text { find: }(a) f(\sqrt{5}) \quad \text { (b) } f(0)$$

Short Answer

Expert verified
(a) \( f(\sqrt{5}) = \frac{1}{2} \), (b) \( f(0) \) is undefined.

Step by step solution

01

Identifying the Function

The function provided is a logarithmic function, specifically, \( f(x) = \log_{5}{x} \), which means the logarithm of \( x \) with base \( 5 \). This function will be used to find the values of \( f(\sqrt{5}) \) and \( f(0) \).
02

Calculating \( f(\sqrt{5}) \)

To calculate \( f(\sqrt{5}) \), substitute \( x = \sqrt{5} \) into the function: \( f(\sqrt{5}) = \log_{5}{(\sqrt{5})} \). To solve, express \( \sqrt{5} \) as \( 5^{1/2} \), thus the expression becomes \( \log_{5}(5^{1/2}) \). Using the logarithmic identity \( \log_{b}(b^a) = a \), it simplifies to \( \frac{1}{2} \).
03

Understanding \( f(0) \)

Consider the function \( f(0) = \log_{5}(0) \). A logarithm is undefined for zero and negative numbers in real numbers. Hence, \( \log_{5}(0) \) is undefined because you cannot raise 5 to any real number to get 0.

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Key Concepts

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

Logarithmic Functions
Logarithmic functions are a fundamental concept in mathematics, especially useful in solving equations where the variable is an exponent. A logarithmic function can be expressed in the form \( f(x) = \log_b(x) \), where \( b \) is the base of the logarithm. This relationship answers the question: "To what power must \( b \) be raised, to obtain \( x \)?"
  • The function \( f(x) = \log_b(x) \) allows us to rearrange exponential equations for better manipulation and understanding.
  • Logarithmic functions are the inverse of exponential functions. If \( f(x) = \log_b(x) \), then this is the opposite of \( g(x) = b^x \).
  • They are defined for positive real numbers, meaning \( x > 0 \) and \( b > 0 \), provided \( b eq 1 \).
Logarithms with a base \( b > 1 \) present a graph that increases as \( x \) increases. In contrast, bases between 0 and 1 yield a decreasing graph. Understanding this framework allows one to effectively solve both simple and complex mathematical problems.
Base of Logarithms
In the context of logarithms, the 'base' is a crucial component defining the logarithmic function's characteristics. For \( \log_b(x) \), \( b \) is the base and must adhere to specific rules for the function to be valid.
  • The base \( b \) must always be a positive number, greater than zero.
  • It's important that \( b eq 1 \); otherwise, the function does not exhibit any meaningful output as raising 1 to any power will always be 1.
  • Common bases used are 10 (common logarithm), \( e \) (natural logarithm), and 2 (binary logarithm).
The choice of base determines the function's rate of growth or decay. Changing the base alters how we interpret the logarithm's result. For instance, in computing, base 2 is prevalent due to its binary system correlation, whereas natural logarithms are often used in continuous growth scenarios due to "e" being a natural growth factor.
Undefined Logarithm
An undefined logarithm occurs when we attempt to evaluate the logarithm for a zero or negative number within the real number system. In the exercise \( f(0) = \log_5(0) \), \( \log_5(0) \) is undefined. This is due to mathematical limitations inherent within logarithms:
  • A positive base raised to any real power can never result in zero or a negative value.
  • Trying to calculate \( \log_b(0) \) implies finding a power that when the base \( b \) is raised to it, equals zero. This is impossible - no real exponent will ever make a positive number zero.
  • Similarly, \( \log_b(x) \) with \( x < 0 \) is undefined because a positive number raised to any real power can't yield a negative number.
The understanding of where logarithms become undefined is pivotal, as it dictates the domain of logarithmic functions, ensuring calculations remain within valid realms. Recognizing when and why a logarithm is undefined aids in avoiding errors when solving problems involving logarithms.

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