Some logarithmic stuff

Algebra Level 2

Positive reals a a and b b are such that a b 3 = 8 a b^3=8 . What is the value of log ⁡ 2 a + 3 log ⁡ 2 b \log _2 a +3\log _2b ?

6 8 2 3

Linkin Duck by Linkin Duck , 33, Vietnam

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1 solution

Chew-Seong Cheong
Jun 27, 2019

a b 3 = 8 log ⁡ 2 ( a b 3 ) = log ⁡ 2 ( 2 3 ) log ⁡ 2 a + log ⁡ 2 ( b 3 ) = 3 log ⁡ 2 a + 3 log ⁡ 2 b = 3 \begin{aligned} ab^3 & = 8 \\ \log_2 (ab^3) & = \log_2(2^3) \\ \log_2 a + \log_2(b^3) & = 3 \\ \log_2 a + 3\log_2 b & = \boxed 3 \end{aligned}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

By question i assume the relation is constant. Checking for trivial case a=1,b=2 you get: log2(1)+3log2(2)=x - > 0+3 * 1 = x = 3

Eric Scholz - 1 year, 11 months ago

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