Answer:
[tex] \log_2(21) = \log_2(3) + \log_2(7) [/tex]
Step-by-step explanation:
To expand the logarithm [tex] \log_2(21) [/tex], we'll need to use the properties of logarithms. In this case, we will apply the product rule:
The product rule states that:
[tex]\Large\boxed{\boxed{ \log_b(a \cdot c) = \log_b(a) + \log_b(c) }}[/tex].
We need to express [tex]21[/tex] as a product of numbers that can be expressed in terms of [tex]2[/tex], since the base of the logarithm is [tex]2[/tex].
We know that [tex]21 = 3 \times 7[/tex].
Thus, we can rewrite [tex]21[/tex] as [tex]3 \times 7[/tex], and then apply the product rule:
[tex] \log_2(21) = \log_2(3 \times 7) [/tex]
Now, using the product rule:
[tex] \log_2(21) = \log_2(3) + \log_2(7) [/tex]
So, the expanded form of [tex] \log_2(21) [/tex] using the product rule is:
[tex] \log_2(3) + \log_2(7) [/tex]
