How do you solve $ {10^x} = 75 $ ?

Hint: Use logarithm to solve this problem because we cannot solve this problem in a conventional method. So, use logarithm to solve this problem. It is one of the complicated problems to deal with. Here $ {10^x} $ is the hint, which indirectly tells us to use logarithm.

Complete step-by-step answer:
Let’s consider the given problem,
 $ {10^x} = 75 $
Taking $ \log $ on both sides we get,
 $ \log {10^x} = \log 75 $
And we know that in logarithm, the power value can be brought as the multiples of the base number i.e.., $ \log {2^x} $ can be written as $ x\log 2 $ , applying this to the given equation we get,
 $ x\log 10 = \log 75 $
Substituting the value $ \log 10 = 1 $ in above equation we get,
 $ x = \log 75 $
Substituting the value for $ \log 75 = 1.8750 $ in above equation,
 $ x = 1.8750 $
This is our required solution.
So, the correct answer is “ $ x = 1.8750 $ ”.

Note: As I mentioned already it is one of the complicated problems in mathematics, there are also some uncomplicated problems in this concept. For instance, let us consider a problem $ {5^x} = 625 $ , here we need to convert $ 625 $ in terms of $ 5 $ to the power of some number. Here if we put $ {5^4} $ , we will get the value $ 625 $ . Hence the problem becomes, $ {5^x} = {5^4} $ , if the bases are the same, we can equate the powers and hence $ x $ is equal to $ 4 $ .