Answer :
To determine which conditions make the function [tex]f[/tex] a logarithmic function, we need to understand the essential properties of logarithmic functions. A logarithmic function generally has a characteristic form [tex]f(x) = k \log_b(x)[/tex] for some base [tex]b[/tex] and constant [tex]k[/tex]. For [tex]f[/tex] to be logarithmic in this scenario, certain relationships between the values of [tex]x[/tex] and their corresponding function values [tex]f(x)[/tex] need to hold.
We observe that:
- [tex]f(a) = k[/tex]
- [tex]f(b) = 2k[/tex]
- [tex]f(c) = 3k[/tex]
- [tex]f(d) = 4k[/tex]
Since logarithmic functions grow proportionally with respect to their inputs, let's analyze the conditions given in the choices:
A) [tex]b = a + 2, c = a + 3, d = a + 4[/tex]
This choice suggests a linear increment, which does not follow the exponential pattern needed for logarithmic growth.
B) [tex]b = 2a, c = 3a, d = 4a[/tex]
This option indicates a multiplicative increase of the values of [tex]x[/tex] corresponding to each function output, which suggests a structure aligned with logarithmic functions, as logarithms scale with products of their bases.
C) [tex]b = a^2, c = a^3, d = a^4[/tex]
This choice indicates an exponential relationship of the inputs relative to the power of [tex]a[/tex], which would not fulfill linear growth characteristic of logarithmic functions.
D) [tex]b = a^2, c = a^4, d = a^8[/tex]
Similar to option C, this suggests a steep growth pattern but does not align with the logarithmic nature which grows with multiplicative constants rather than powers directly.
Upon evaluating each of these choices, it becomes evident that option B satisfies the requirements for [tex]f[/tex] to be logarithmic. Therefore, the chosen answer is:
B) b = 2a, c = 3a, and d = 4a.
The complete question is shown below.
To determine if [tex]\( f(x) \)[/tex] is a logarithmic function based on the values provided in the table, let's break down the requirements step-by-step.
Given:
- Four points with [tex]\( x \)[/tex] values: [tex]\( a, b, c, d \)[/tex] where [tex]\( 1 < a < b < c < d \)[/tex]
- Corresponding [tex]\( f(x) \)[/tex] values: [tex]\( k, 2k, 3k, 4k \)[/tex] where [tex]\( k > 1 \)[/tex]
For [tex]\( f(x) \)[/tex] to be a logarithmic function, it should represent a relationship like:
[tex]\[ f(x) = k \cdot \log_b(x) \][/tex]
Here’s a way to check if [tex]\( f(x) \)[/tex] might actually be a logarithmic function:
1. Logarithmic Properties: If [tex]\( f(x) \)[/tex] is logarithmic, the function has a property that when you increase the argument [tex]\( x \)[/tex] in a multiplicative fashion, the function value differences will form a constant additive pattern.
2. Differences in [tex]\( f(x) \)[/tex] Values:
- If [tex]\( f(x) \)[/tex] is logarithmic, then changes in [tex]\( x \)[/tex] should result in consistent increments in [tex]\( f(x) \)[/tex].
- Given the values are [tex]\( k, 2k, 3k, 4k \)[/tex], the differences among successive [tex]\( f(x) \)[/tex] values are all equal to [tex]\( k \)[/tex].
3. Checking Consistency with Logarithms:
- The condition that ensures [tex]\( f(x) \)[/tex] is logarithmic is that the ratios of consecutive function values should be consistent.
- Compute the ratios:
- [tex]\( f(b)/f(a) = (2k)/k = 2 \)[/tex]
- [tex]\( f(c)/f(b) = (3k)/(2k) = \frac{3}{2} \)[/tex]
- [tex]\( f(d)/f(c) = (4k)/(3k) = \frac{4}{3} \)[/tex]
- These ratios ([tex]\( 2, \frac{3}{2}, \frac{4}{3} \)[/tex]) show a consistent pattern of multiplication that corresponds with a logarithmic model under certain conditions.
Thus, the condition to ensure that [tex]\( f(x) \)[/tex] behaves as a logarithmic function with the patterns of [tex]\( f(x) = \log(x) \cdot k \)[/tex] would require that these multiplicative ratios are consistently constant.
Since all conditions of increasing constant multiplicative differences are met, [tex]\( f(x) \)[/tex] can indeed model a logarithmic function. However, adjusting base [tex]\( b \)[/tex] or scaling may need to align with specific differences, based on real data context.
Given:
- Four points with [tex]\( x \)[/tex] values: [tex]\( a, b, c, d \)[/tex] where [tex]\( 1 < a < b < c < d \)[/tex]
- Corresponding [tex]\( f(x) \)[/tex] values: [tex]\( k, 2k, 3k, 4k \)[/tex] where [tex]\( k > 1 \)[/tex]
For [tex]\( f(x) \)[/tex] to be a logarithmic function, it should represent a relationship like:
[tex]\[ f(x) = k \cdot \log_b(x) \][/tex]
Here’s a way to check if [tex]\( f(x) \)[/tex] might actually be a logarithmic function:
1. Logarithmic Properties: If [tex]\( f(x) \)[/tex] is logarithmic, the function has a property that when you increase the argument [tex]\( x \)[/tex] in a multiplicative fashion, the function value differences will form a constant additive pattern.
2. Differences in [tex]\( f(x) \)[/tex] Values:
- If [tex]\( f(x) \)[/tex] is logarithmic, then changes in [tex]\( x \)[/tex] should result in consistent increments in [tex]\( f(x) \)[/tex].
- Given the values are [tex]\( k, 2k, 3k, 4k \)[/tex], the differences among successive [tex]\( f(x) \)[/tex] values are all equal to [tex]\( k \)[/tex].
3. Checking Consistency with Logarithms:
- The condition that ensures [tex]\( f(x) \)[/tex] is logarithmic is that the ratios of consecutive function values should be consistent.
- Compute the ratios:
- [tex]\( f(b)/f(a) = (2k)/k = 2 \)[/tex]
- [tex]\( f(c)/f(b) = (3k)/(2k) = \frac{3}{2} \)[/tex]
- [tex]\( f(d)/f(c) = (4k)/(3k) = \frac{4}{3} \)[/tex]
- These ratios ([tex]\( 2, \frac{3}{2}, \frac{4}{3} \)[/tex]) show a consistent pattern of multiplication that corresponds with a logarithmic model under certain conditions.
Thus, the condition to ensure that [tex]\( f(x) \)[/tex] behaves as a logarithmic function with the patterns of [tex]\( f(x) = \log(x) \cdot k \)[/tex] would require that these multiplicative ratios are consistently constant.
Since all conditions of increasing constant multiplicative differences are met, [tex]\( f(x) \)[/tex] can indeed model a logarithmic function. However, adjusting base [tex]\( b \)[/tex] or scaling may need to align with specific differences, based on real data context.
