Which of the following uses the properties of logarithms to correctly rewrite f(x)=log2 16x^3

Question

?(1 point)
Responses

f(x)=3log2 x+4
f left parenthesis x right parenthesis equals 3 logarithm subscript 2 baseline x plus 4

f(x)=log2 3x+4
f left parenthesis x right parenthesis equals logarithm subscript 2 baseline 3 x plus 4

f(x)=3log2 x+12
f left parenthesis x right parenthesis equals 3 logarithm subscript 2 baseline x plus 12

f(x)=16log2 x^3
f left parenthesis x right parenthesis equals 16 logarithm subscript 2 baseline x cubed

Answer 1

To rewrite the function \( f(x) = \log_2 (16x^3) \) using the properties of logarithms, we can apply the product property and the power property.

1. Product Property: \( \log_b (mn) = \log_b m + \log_b n \)
2. Power Property: \( \log_b (m^n) = n \log_b m \)

Using these properties, we can rewrite \( f(x) \):

\[ f(x) = \log_2 (16x^3) = \log_2 16 + \log_2 (x^3) \]

Next, we apply the power property:

\[ \log_2 (x^3) = 3 \log_2 x \]

Now we also need to compute \( \log_2 16 \):

Since \( 16 = 2^4 \): \[ \log_2 16 = 4 \]

Thus we have:

\[ f(x) = \log_2 16 + 3 \log_2 x = 4 + 3 \log_2 x \]

So the corrected form of \( f(x) \) is:

\[ f(x) = 3 \log_2 x + 4 \]

Among the given options, the correct choice is:

f(x) = 3 log₂ x + 4.