Question

Solve the equation log5(x−11)+log5(x−6)=2 X is equal to

Answer 1

Answer:

Step-by-step explanation:

We have,

$\sf{\log_{5}(x-11)+\log_{5}(x-6)=2}$

domain: x > 6

$\sf{\implies\log_{5}(x-11)(x-6)=2}$

$\sf{\implies(x-11)(x-6)=5^2}$

$\sf{\implies\,x^2-17x+66=25}$

$\sf{\implies\,x^2-17x+66-25=0}$

$\sf{\implies\,x^2-17x+41=0}$

$\sf{\implies\,x=\dfrac{-(-17)\pm\sqrt{(-17)^2-4\cdot(1)\cdot(41)}}{2}}$

$\sf{\implies\,x=\dfrac{17\pm\sqrt{289-164}}{2}}$

$\sf{\implies\,x=\dfrac{17\pm\sqrt{125}}{2}}$

$\sf{\implies\,x=\dfrac{17+5\sqrt{5}}{2}\,\,\,\,or\,\,\,\,x=\dfrac{17-5\sqrt{5}}{2}}$

$\sf{But\,\,\, x\ne\dfrac{17-5\sqrt{5}}{2},\,\,\,as\,\,\dfrac{17-5\sqrt{5}}{2}<6}$

So, only solution is $\sf{x=\dfrac{17+5\sqrt{5}}{2}}$