Answered

Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.

Given:

[tex]\[ f(x) = \log_2(x) \][/tex]

Determine [tex]\( g(x) \)[/tex].

Which of the following expressions is equal to [tex]\( g(x) \)[/tex]?

A. [tex]\( \frac{1}{2} \cdot \log_2(x) \)[/tex]

B. [tex]\( \log_2\left(\frac{1}{2} x\right) \)[/tex]

C. [tex]\( \log_2(2x) \)[/tex]

D. [tex]\( 2 \cdot \log_2(x) \)[/tex]

Sagot :

GinnyAnswer
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]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.