To determine which expression for [tex]\( g(x) \)[/tex] is equivalent to a function [tex]\( f(x) = \log_2(x) \)[/tex], we can evaluate each given option by understanding properties of logarithms.
### Properties of Logarithms
1. Logarithm of a Product: [tex]\(\log_b(xy) = \log_b(x) + \log_b(y)\)[/tex]
2. Logarithm of a Power: [tex]\(\log_b(x^a) = a \log_b(x)\)[/tex]
3. Logarithm of a Quotient: [tex]\(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)[/tex]
Let's evaluate each option one by one:
Option A: [tex]\(\frac{1}{2} \cdot \log_2(x)\)[/tex]
This option suggests that the output of the logarithmic function is scaled by a factor of [tex]\(\frac{1}{2}\)[/tex]. It could be written as:
[tex]\[ g(x) = \frac{1}{2} \log_2(x) \][/tex]
Option B: [tex]\(\log_2\left(\frac{1}{2} x\right)\)[/tex]
Using the property of the logarithm of a product, we can rewrite:
[tex]\[ \log_2\left(\frac{1}{2} x\right) = \log_2\left(\frac{1}{2}\right) + \log_2(x) \][/tex]
Since [tex]\(\log_2\left(\frac{1}{2}\right) = \log_2(1) - \log_2(2)\)[/tex], and knowing [tex]\(\log_2(2) = 1\)[/tex]:
[tex]\[ \log_2\left(\frac{1}{2}\right) = -1 \][/tex]
So, this simplifies to:
[tex]\[ \log_2\left(\frac{1}{2} x\right) = -1 + \log_2(x) \][/tex]
Option C: [tex]\(\log_2(2 x)\)[/tex]
Again using the property of the logarithm of a product:
[tex]\[ \log_2(2 x) = \log_2(2) + \log_2(x) \][/tex]
Since [tex]\(\log_2(2) = 1\)[/tex]:
[tex]\[ \log_2(2 x) = 1 + \log_2(x) \][/tex]
Option D: [tex]\(2 \cdot \log_2(x)\)[/tex]
This option scales the logarithm by a factor of [tex]\(2\)[/tex]:
[tex]\[ g(x) = 2 \log_2(x) \][/tex]
### Conclusion
To determine which option is equivalent to [tex]\( g(x) \)[/tex], we compare the resultant forms:
- Option A: [tex]\(\frac{1}{2} \log_2(x)\)[/tex]
- Option B: [tex]\(-1 + \log_2(x)\)[/tex]
- Option C: [tex]\(1 + \log_2(x)\)[/tex]
- Option D: [tex]\(2 \log_2(x)\)[/tex]
The expression that matches the form [tex]\( g(x) = 1 + \log_2(x) \)[/tex] is derived from Option C:
[tex]\[ \log_2(2 x) \][/tex]
Thus, the expression [tex]\( g(x) = \log_2(2 x) \)[/tex] is the correct one.
The correct answer is:
Option C: [tex]\(\log_2(2 x)\)[/tex]
