Question

Solve: $ {3^x} = {5^{x - 2}} $

Answer 1

Hint: Here, as x is in exponent, so apply logarithm on both sides. Simplify the equation obtained using logarithm formulae. At the last simplify the value of x in terms of log or number.

Complete step-by-step answer:
We have, $ {3^x} = {5^{x - 2}} $
Taking log on both sides,
$ \log \left( {{3^x}} \right) = \log \left( {{5^{x - 2}}} \right) $
$x log 3 = (x – 2) log 5$ as $ \left[ {\log {a^x} = x\log a} \right] $
$ x log 3 = x log 5 – 2 log 5$
Rearranging the terms
$2 log 5 = x log 5 − x log 3 $
Taking x common from both the terms containing x
$2 log 5 = x(log 5 − log 3) $
 $ \Rightarrow \log {5^2} = x\log \dfrac{5}{3} $ [log a − log b = log (a/b)]
 $ \Rightarrow 2\log 5 = x\dfrac{{\log 5}}{{\log 3}} $ [As base number of all log are same]
 $ \Rightarrow \dfrac{{2\log 5 \times \log 3}}{{\log 5}} = x $
On simplifying
 $ \Rightarrow x = 2\log 3 $
So, the correct answer is “ $ x = 2\log 3 $ ”.

Note: Take all basis of logarithm as same, either base e which is natural base or take base as 10. For simple questions, we can use the hit and trial method, in which we put different values of x and check whether the equation is satisfied or not. If the taken value of x satisfies the equation then that value can be taken as result. But the hit and trial method is useful for simple problems, and cannot be used for complicated problems. Also hit and trial method is quite lengthy for some cases and we cannot say whether value obtained is the only solution of the equation or not.