Which of the following uses the properties of logarithms to correctly rewrite f(x)=log216x3

Question

?(1 point)
Responses

f(x)=16log2x3
f left parenthesis x right parenthesis equals 16 logarithm subscript 2 baseline x cubed

f(x)=log23x+4
f left parenthesis x right parenthesis equals logarithm subscript 2 baseline 3 x plus 4

f(x)=3log2x+4
f left parenthesis x right parenthesis equals 3 logarithm subscript 2 baseline x plus 4

f(x)=3log2x+12
f left parenthesis x right parenthesis equals 3 logarithm subscript 2 baseline x plus 12

Answer 1

To rewrite the function \( f(x) = \log_{2}(16x^{3}) \) using the properties of logarithms, we can use the following properties:

1. \( \log_{b}(mn) = \log_{b}m + \log_{b}n \)
2. \( \log_{b}(m^n) = n \cdot \log_{b}m \)

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

\[ f(x) = \log_{2}(16x^{3}) = \log_{2}(16) + \log_{2}(x^{3}) \]

Next, we compute \( \log_{2}(16) \), since \( 16 = 2^4 \):

\[ \log_{2}(16) = \log_{2}(2^4) = 4 \]

Now, applying the second property to \( \log_{2}(x^{3}) \):

\[ \log_{2}(x^{3}) = 3 \log_{2}(x) \]

Now we can substitute back into our function:

\[ f(x) = 4 + 3 \log_{2}(x) \]

This can be rearranged as:

\[ f(x) = 3 \log_{2}(x) + 4 \]

So, the correct expression that uses the properties of logarithms to rewrite \( f(x) \) is:

\[ f(x) = 3\log_{2}(x) + 4 \]

Thus, the correct response is:

f(x) = 3log2x + 4.