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.