Evaluate log base 4 of 2+ log base 4 of 128

Problem

(log_4)(2)+(log_4)(128)

Solution

1. Apply the product rule for logarithms, which states that (log_b)(m)+(log_b)(n)=(log_b)(m⋅n)

(log_4)(2)+(log_4)(128)=(log_4)(2⋅128)

2. Multiply the numbers inside the logarithm.

2⋅128=256

(log_4)(256)

3. Rewrite the argument as a power of the base. Since 4^4=256 we can substitute 256 with 4^4

(log_4)(4^4)

4. Simplify using the property (log_b)(b^x)=x

(log_4)(4^4)=4

Final Answer

(log_4)(2)+(log_4)(128)=4

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