Expand the Logarithmic Expression log base 2 of 20

Problem

(log_2)(20)

Solution

1. Identify the factors of the argument 20 that include powers of the base 2.

20=4⋅5

20=2⋅5

2. Apply the product rule for logarithms, which states (log_b)(x*y)=(log_b)(x)+(log_b)(y)

(log_2)(2⋅5)=(log_2)(2)+(log_2)(5)

3. Apply the power rule for logarithms, which states (log_b)(xp)=p*(log_b)(x)

(log_2)(2)=2*(log_2)(2)

4. Simplify the expression using the property (log_b)(b)=1

2*(log_2)(2)=2*(1)

2*(log_2)(2)=2

5. Combine the simplified terms to reach the final expanded form.

2+(log_2)(5)

Final Answer

(log_2)(20)=2+(log_2)(5)

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