The Power of Logs

Algebra Level 2

40000 x 10 log ⁡ 4 40000 = 1 \large 40000 x ^ {10\log_4 40000} = 1

The real value x = 8 − a b x = 8^{- \frac{a}{b}} , where a a and b b are coprime positive integers, satisfies the equation above. What is the value of a + b a + b ?


The answer is 16.

Chris Sapiano by Chris Sapiano , 19, United Kingdom

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2 solutions

40000 x 10 log ⁡ 4 40000 = 1 Taking log ⁡ 4 on both sides log ⁡ 4 40000 + 10 log ⁡ 4 40000 log ⁡ 4 x = 0 log ⁡ 4 x = − log ⁡ 4 40000 10 log ⁡ 4 40000 = − 1 10 ⟹ x = 4 − 1 10 = 2 − 1 5 = 8 − 1 15 \begin{aligned} 40000 x^{10\log_4 40000} & = 1 & \small \color{#3D99F6} \text{Taking }\log_4 \text{ on both sides} \\ \log_4 40000 + 10 \log_4 40000 \log_4 x & = 0 \\ \log_4 x & = - \frac {\log_4 40000}{10 \log_4 40000} = - \frac 1{10} \\ \implies x & = 4^{-\frac 1{10}} = 2^{-\frac 15} = 8^{-\frac 1{15}} \end{aligned}

Therefore, a + b = 1 + 15 = 16 a+b = 1 + 15 = \boxed{16} .

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40000 ( 8 − a b ) 10 log ⁡ 4 ( 40000 ) = 1 ⇒ a b = 1 15 40000 \left(8^{-\frac{a}{b}}\right)^{10 \log _4(40000)}=1 \Rightarrow \frac{a}{b}=\frac{1}{15}

40000 ( 8 − 1 15 ) 10 log ⁡ 4 ( 40000 ) = 1 40000 \left(8^{-\frac{1}{15}}\right)^{10 \log _4(40000)} = 1

0 Helpful 0 Interesting 0 Brilliant 2 Confused

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