Answer :
To analyze the logarithmic function [tex]\( f(x) \)[/tex] given in the table and determine its domain and range, let's start by examining the values in the table.
We are given the following pairs of [tex]\( (x, y) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline \frac{1}{125} & -3 \\ \hline \frac{1}{25} & -2 \\ \hline \frac{1}{5} & -1 \\ \hline 1 & 0 \\ \hline 5 & 1 \\ \hline 25 & 2 \\ \hline 125 & 3 \\ \hline \end{array} \][/tex]
### Step-by-Step Analysis:
1. Identify the Function Form:
- Notice that as [tex]\( x \)[/tex] increases by factors of 5, the value of [tex]\( y \)[/tex] increases by 1. Specifically:
- [tex]\( 5^(-3) = \frac{1}{125} \)[/tex]
- [tex]\( 5^(-2) = \frac{1}{25} \)[/tex]
- [tex]\( 5^(-1) = \frac{1}{5} \)[/tex]
- [tex]\( 5^0 = 1 \)[/tex]
- [tex]\( 5^1 = 5 \)[/tex]
- [tex]\( 5^2 = 25 \)[/tex]
- [tex]\( 5^3 = 125 \)[/tex]
- This pattern suggests that the function can be written as:
[tex]\[ y = \log_5(x) \][/tex]
2. Confirming the Function:
- Verify the behavior of the function with the given pairs:
[tex]\[ \begin{align*} \log_5\left(\frac{1}{125}\right) &= -3, \\ \log_5\left(\frac{1}{25}\right) &= -2, \\ \log_5\left(\frac{1}{5}\right) &= -1, \\ \log_5(1) &= 0, \\ \log_5(5) &= 1, \\ \log_5(25) &= 2, \\ \log_5(125) &= 3. \end{align*} \][/tex]
- As each [tex]\( x \)[/tex] value can be expressed as [tex]\( 5 \)[/tex] raised to some power, the corresponding [tex]\( y \)[/tex] values match, confirming that we have [tex]\( y = \log_5(x) \)[/tex].
3. Plotting the Function:
- Using the pairs [tex]\((x, y)\)[/tex], we plot the points:
[tex]\[ (0.008, -3), (0.04, -2), (0.2, -1), (1, 0), (5, 1), (25, 2), (125, 3) \][/tex]
- The graph of [tex]\( y = \log_5(x) \)[/tex] passes through these points.
4. Determining the Domain:
- For a logarithmic function [tex]\( y = \log_b(x) \)[/tex], the domain is [tex]\( x > 0 \)[/tex] because a logarithm is only defined for positive values.
- Here, the domain is:
[tex]\[ (0, \infty) \quad \text{in interval notation} \][/tex]
5. Determining the Range:
- The range of a logarithmic function is all real numbers since the output (logarithm) can be any real number.
- Therefore, the range is:
[tex]\[ (-\infty, \infty) \quad \text{in interval notation} \][/tex]
### Summary:
- Function:
[tex]\[ f(x) = \log_5(x) \][/tex]
- Domain:
[tex]\[ (0, \infty) \][/tex]
- Range:
[tex]\[ (-\infty, \infty) \][/tex]
We are given the following pairs of [tex]\( (x, y) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline \frac{1}{125} & -3 \\ \hline \frac{1}{25} & -2 \\ \hline \frac{1}{5} & -1 \\ \hline 1 & 0 \\ \hline 5 & 1 \\ \hline 25 & 2 \\ \hline 125 & 3 \\ \hline \end{array} \][/tex]
### Step-by-Step Analysis:
1. Identify the Function Form:
- Notice that as [tex]\( x \)[/tex] increases by factors of 5, the value of [tex]\( y \)[/tex] increases by 1. Specifically:
- [tex]\( 5^(-3) = \frac{1}{125} \)[/tex]
- [tex]\( 5^(-2) = \frac{1}{25} \)[/tex]
- [tex]\( 5^(-1) = \frac{1}{5} \)[/tex]
- [tex]\( 5^0 = 1 \)[/tex]
- [tex]\( 5^1 = 5 \)[/tex]
- [tex]\( 5^2 = 25 \)[/tex]
- [tex]\( 5^3 = 125 \)[/tex]
- This pattern suggests that the function can be written as:
[tex]\[ y = \log_5(x) \][/tex]
2. Confirming the Function:
- Verify the behavior of the function with the given pairs:
[tex]\[ \begin{align*} \log_5\left(\frac{1}{125}\right) &= -3, \\ \log_5\left(\frac{1}{25}\right) &= -2, \\ \log_5\left(\frac{1}{5}\right) &= -1, \\ \log_5(1) &= 0, \\ \log_5(5) &= 1, \\ \log_5(25) &= 2, \\ \log_5(125) &= 3. \end{align*} \][/tex]
- As each [tex]\( x \)[/tex] value can be expressed as [tex]\( 5 \)[/tex] raised to some power, the corresponding [tex]\( y \)[/tex] values match, confirming that we have [tex]\( y = \log_5(x) \)[/tex].
3. Plotting the Function:
- Using the pairs [tex]\((x, y)\)[/tex], we plot the points:
[tex]\[ (0.008, -3), (0.04, -2), (0.2, -1), (1, 0), (5, 1), (25, 2), (125, 3) \][/tex]
- The graph of [tex]\( y = \log_5(x) \)[/tex] passes through these points.
4. Determining the Domain:
- For a logarithmic function [tex]\( y = \log_b(x) \)[/tex], the domain is [tex]\( x > 0 \)[/tex] because a logarithm is only defined for positive values.
- Here, the domain is:
[tex]\[ (0, \infty) \quad \text{in interval notation} \][/tex]
5. Determining the Range:
- The range of a logarithmic function is all real numbers since the output (logarithm) can be any real number.
- Therefore, the range is:
[tex]\[ (-\infty, \infty) \quad \text{in interval notation} \][/tex]
### Summary:
- Function:
[tex]\[ f(x) = \log_5(x) \][/tex]
- Domain:
[tex]\[ (0, \infty) \][/tex]
- Range:
[tex]\[ (-\infty, \infty) \][/tex]
