Question

5^x+5^(x-2) = 650, find the value of x

Answer 1

SOLUTION

TO DETERMINE

The value of x when

$\displaystyle \sf{ {5}^{x} + {5}^{(x - 2)} = 650}$

EVALUATION

$\displaystyle \sf{ {5}^{x} + {5}^{(x - 2)} = 650}$

$\displaystyle \sf{ \implies {5}^{x} + \frac{ {5}^{x} }{ {5}^{2} } = 650}$

$\displaystyle \sf{ Let \: \: \: a = {5}^{x} }$

From above we get

$\displaystyle \sf{ a + \frac{ a }{ 25 } = 650}$

$\displaystyle \sf{ \implies \frac{25 a + a }{ 25 } = 650}$

$\displaystyle \sf{ \implies \frac{26 a }{ 25 } = 650}$

$\displaystyle \sf{ \implies a = 650 \times \frac{25}{26} }$

$\displaystyle \sf{ \implies {5}^{x} = 25 \times 25}$

$\displaystyle \sf{ \implies {5}^{x} = {25}^{2} }$

$\displaystyle \sf{ \implies {5}^{x} = {5}^{4} }$

$\displaystyle \sf{ \implies x = 4 }$

FINAL ANSWER

Hence the required solution is x = 4

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