If-log-10-ax-log-10-bx-1-find-x-in-term-a-and-b-

Question Number 80844 by jagoll last updated on 07/Feb/20

I f (\log_{10} (a x)) (\log_{10} (b x)) = - 1

f i n d x i n t e r m a a n d b

Commented by jagoll last updated on 07/Feb/20

c a n a n y o n e h e l p m e

Commented by jagoll last updated on 07/Feb/20

\frac{b}{a} ⩾ \dots

i t t h e a n s w e r

Answered by mr W last updated on 07/Feb/20

s o l u t i o n e x i s t s o n l y f o r a, b > 0 o r a, b < 0 .

a s s u m e a, b > 0 i n f o l l o w i n g .

\log (a x) \times \log (b x) = - 1

(\log a + \log x) (\log b + \log x) = - 1

(A + X) (B + X) = - 1

X^{2} + (A + B) X + (A B + 1) = 0

X = \frac{- (A + B) \pm \sqrt{{(A - B)}^{2} - 4}}{2}

\log x = \frac{- (\log a + \log b) \pm \sqrt{{(\log a - \log b)}^{2} - 4}}{2}

\log x = \frac{- \log a b \pm \sqrt{{(\log \frac{a}{b})}^{2} - 4}}{2}

\Rightarrow x = 10^{\frac{- \log a b \pm \sqrt{{(\log \frac{a}{b})}^{2} - 4}}{2}}

{(\log \frac{a}{b})}^{2} ⩾ 4

\log \frac{a}{b} ⩽ - 2 o r ⩾ 2

\Rightarrow 0 < \frac{a}{b} ⩽ \frac{1}{100} o r \frac{a}{b} ⩾ 100

Commented by jagoll last updated on 07/Feb/20

t h a n k y o u m i s t e r W

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