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Chapter 4: Problem 98

Fill in the missing coordinate in each ordered pair so that the pair is a solution to the given equation. $$y=10^{x}$$ $$(3, \quad),(\quad, 1),(-1, \quad),(\quad, 0.01)$$

Short Answer

Expert verified
(3, 1000), (0, 1), (-1, 0.1), (-2, 0.01)

Step by step solution

01

Understand the Given Equation

The given equation is a logarithmic function given by \( y = 10^x \). In this scenario, the goal is to fill in the missing coordinates for the ordered pairs so that they satisfy this equation.
02

Solve for the First Pair (3, \, \_)

Substitute \( x = 3 \) in the equation \( y = 10^x \). This gives: \( y = 10^3 \) Simplifying, \( y = 1000 \). Hence, the first ordered pair is (3, 1000).
03

Solve for the Second Pair (\_, 1)

Here, \( y = 1 \). Substitute \( y = 1 \) into the equation \( y = 10^x \), we get: \( 1 = 10^x \). Solving for \( x \), we see that \( x = 0 \). Hence, the second ordered pair is (0, 1).
04

Solve for the Third Pair (-1, \_)

Substitute \( x = -1 \) into the equation \( y = 10^{-1} \), which simplifies to: \( y = 0.1 \). Thus, the third ordered pair is (-1, 0.1).
05

Solve for the Fourth Pair (\_, 0.01)

Here, \( y = 0.01 \). Substitute \( y = 0.01 \) into the equation \( y = 10^x \), giving: \( 0.01 = 10^x \). By understanding that \( 0.01 = 10^{-2} \), we conclude \( x = -2 \). Hence, the fourth ordered pair is (-2, 0.01).

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Key Concepts

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

Exponential Functions
An exponential function is a type of function where a constant base is raised to a variable exponent. In this case, the given equation is \( y = 10^x \). This means that for any value of \( x \), \( y \) will be equal to 10 raised to the power of \( x \).
Exponential functions are widely used in various fields such as science, finance, and engineering. They are characterized by rapid growth or decay.
Here are some key points about exponential functions:
  • \( 10^0 = 1 \), because any number raised to the power of 0 is 1.
  • \( 10^1 = 10 \), the base itself.
  • \( 10^{-1} = 0.1 \), because raising a number to a negative exponent means dividing 1 by that number.
  • \( 10^2 = 100 \), indicating exponential growth.
These properties were used to find the missing coordinates in the given ordered pairs.
Solving Equations
Solving equations involves finding the value of the variable that makes the equation true. In our problem, we are given the equation \( y = 10^x \) and we need to find the missing coordinates for the ordered pairs.
Here is how each step works:
  • For the first pair, \( (3, \_) \), we substitute \( x = 3 \) into the equation, yielding \( y = 10^3 = 1000 \).
  • For the second pair, \( (\_, 1) \), we know that \( y = 1 \). Setting \( 10^x = 1 \), we infer that \( x = 0 \) because any number to the power of zero is 1.
  • For the third pair, \( (-1, \_) \), substituting \( x = -1 \) gives \( y = 10^{-1} = 0.1 \).
  • For the fourth pair, \( (\_, 0.01) \), knowing \( y = 0.01 \), we recognize that \( 0.01 = 10^{-2} \), so \( x = -2 \).
By systematically replacing \( x \) or \( y \) and solving for the missing variable, we find the required values.
Ordered Pairs
Ordered pairs are a fundamental component of coordinate geometry, typically written as \( (x, y) \) where \( x \) is the first element and \( y \) is the second. They are used to denote points on a coordinate plane.
For the given problem, each ordered pair needs to satisfy the equation \( y = 10^x \). Here鈥檚 a quick breakdown:
  • In the pair \( (3, 1000) \), when \( x = 3 \), \( y \) is 1000, aligning with \( y = 10^3 \).
  • For \( (0, 1) \), \( x = 0 \) and \( y = 1 \), confirming \( y = 10^0 \).
  • In the pair \( (-1, 0.1) \), \( x = -1 \) and \( y \) is 0.1 which matches \( y = 10^{-1} \).
  • Finally, \( (-2, 0.01) \) pairs \( x = -2 \) with \( y = 0.01 \), fitting the equation as \( y = 10^{-2} \).
The concept of ordered pairs helps in plotting and solving equations graphically, as each pair represents a unique point.

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