Point One

Algebra Level 3

If log ⁡ 5 ( x ) = a \log_{5}(x) = a and log ⁡ 2 ( y ) = a \log _{2}(y)=a then find the value of 10 0 2 a − 1 100^{2a-1} in terms of x x and y y .

x 2 y 2 100 \frac{x^2 y^2}{100} x 4 y 4 10 \frac{x^4 y^4}{10} x y 100 \frac{xy}{100} x y 10 \frac{xy}{10} x 4 y 4 100 \frac{x^4 y^4}{100}

Keerthi Reddy by Keerthi Reddy , 45, India

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2 solutions

Chew-Seong Cheong
May 19, 2016

{ log ⁡ 5 x = a ⟹ x = 5 a log ⁡ 2 y = a ⟹ y = 2 a \begin{cases} \log_5 x = a & \implies x = 5^a \\ \log_2 y = a & \implies y = 2^a \end{cases}

Now, we have: 10 0 2 a − 1 = ( 5 2 ⋅ 2 2 ) 2 a − 1 = 5 4 a − 2 ⋅ 2 4 a − 2 = 5 4 a ⋅ 2 4 a 5 2 ⋅ 2 2 = x 4 y 4 100 100^{2a-1} = (5^2\cdot 2^2)^{2a-1} = 5^{4a-2}\cdot 2^{4a-2} = \dfrac{5^{4a}\cdot 2^{4a}}{5^2\cdot 2^2} = \boxed{\dfrac{x^4y^4}{100}}

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Keerthi Reddy
May 19, 2016

5^{a}=x and 2^{a}=y... Multiplying both the equation gives us 10^{a}=xy------- (I)

But we want 100^ {2a}* 100^-1 which implies 10^{4a}* 100^-1-----------------(II) to reach the above statement we need to power the statement (I) by 4 and multiply with 100^-1 if we do so we get, x^4y^4/100

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