Question

5^x + 5^x 5^x +5^x =500

find x​

Answer 1

4(5^x)=500

5^x=125

5^x=5^3

here

bases are equal so powers are also equal

so the value of x=3

Answer 2

$( {5}^{x} ) {}^{2} + 2 ( {5}^{x} ) - 500 = 0$

${5}^{x} = \frac{ - 2 ± \sqrt{2004} }{2} = - 1 ± \sqrt{501}$

Therefore,

$x = log_{5}( - 1 ± \sqrt{501} )$

$\implies x = log_{5}( \sqrt{501} - 1) \\ \implies x = log_{5}( \sqrt{501} + 1 ) + log_{5}(e) i\pi$