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Magazine Chapter 12: Problem 110
What can we conclude about the functions $$y=f(x)=10^{x} \text { and } y=g(x)=\log _{10} x ?$$
Short Answer
Expert verified
The functions \( y = 10^x \) and \( y = \log_{10} x \) are inverses of each other.
Step by step solution
01
Identify the functions
The functions given are: \( y = f(x) = 10^x \) and \( y = g(x) = \log_{10} x \). We need to determine the relationship between these functions.
02
Understand the exponential function
The first function \( y = 10^x \) is an exponential function. Exponential functions grow rapidly and are defined for all real numbers with a domain of \( (-\infty, \infty) \) and a range of \( (0, \infty) \).
03
Understand the logarithmic function
The second function \( y = \log_{10} x \) is a logarithmic function. Logarithmic functions increase slowly and are the inverse of exponential functions. This function is defined only for positive real numbers with a domain of \( (0, \infty) \) and a range of \( (-\infty, \infty) \).
04
Establish the inverse relationship
Recall that exponentiation and logarithms are inverse operations. Therefore, the functions \( y = 10^x \) and \( y = \log_{10} x \) are inverses of each other. This means that \( f(g(x)) = 10^{\log_{10} x} = x \) and \( g(f(x)) = \log_{10}(10^x) = x \).
05
Graphical representation
If we were to graph these functions, we would see that \( y = 10^x \) and \( y = \log_{10} x \) are symmetric with respect to the line \( y = x \). This is a property of inverse functions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
exponential function
An exponential function is a type of mathematical function where a constant base is raised to a variable exponent. In our exercise, the exponential function is given by \( y = 10^x \).
- Exponential functions grow very quickly. This is because as the value of the exponent increases, the value of the function increases exponentially.
- For the function \( y = 10^x \), the base is 10. So, as the value of \( x \) increases, \( 10^x \) grows larger and larger.
- The domain of \( 10^x \) is all real numbers \( (-\infty, \infty) \). This means that you can plug in any real number for \( x \).
- The range of \( 10^x \) is \( (0, \infty) \), meaning the output of the function is always positive and grows from 0 to infinity.
logarithmic function
A logarithmic function is essentially the inverse of an exponential function. It answers the question: 'To what exponent must the base be raised to produce a given number?'. In our exercise, the logarithmic function is given by \( y = \log_{10} x \).
- Logarithmic functions grow slowly compared to exponential functions. As \( x \) grows, \( \log_{10} x \) increases, but much more gradually.
- The base of our logarithmic function is 10, meaning \( y = \log_{10} x \) tells us what exponent 10 must be raised to in order to get \( x \). For example, \( \log_{10} 100 = 2 \) because \( 10^2 = 100 \).
- The domain of \( \log_{10} x \) is \( (0, \infty) \). This means that x must be positive.
- The range of \( \log_{10} x \) is all real numbers \( (-\infty, \infty) \). So, \( \log_{10} x \) can take any real number value.
domain and range
Understanding the domain and range of functions is crucial for solving and graphing them effectively.
- The domain refers to all possible input values (x-values) that the function can accept. For \( y = 10^x \), the domain is \( (-\infty, \infty) \), meaning you can input any real number.
- The range refers to all possible output values (y-values) that the function can produce. For \( y = 10^x \), the range is \( (0, \infty) \), meaning the output is always positive and can be any positive number.
- For \( y = \log_{10} x \), the domain is \( (0, \infty) \), meaning x must be positive. The range for \( y = \log_{10} x \) is all real numbers \( (-\infty, \infty) \), allowing the output to take any real value.
