Question

How do you solve ${{\log }_{x}}44=2?$

Answer 1

Hint: We will use the logarithmic identities to solve the problem. We know the identity ${{\log }_{b}}a=c$ if and only if ${{b}^{c}}=a.$ We will compare the values of the given problem with the identity.

Complete step by step solution:
Let us consider the given problem ${{\log }_{x}}44=2.$
We are asked to solve the given logarithmic equation to find the value of the unknown variable $x.$
For solving the given logarithmic equation, we need to use the logarithmic identities.
According to the situation we are in, we are supposed to choose the logarithmic identity ${{\log }_{b}}a=c$ if and only if ${{b}^{c}}=a.$
We know that we have the similar case as the above identity, because the left-hand side of our equation is similar to the first part of the above written identity.
So, we know that we can compare the equation we have with the above identity in order to find the value of the unknown number $x.$ Then, we will make necessary rearrangements to find the solution of this problem.
When we compare the values, we will get ${{\log }_{b}}a={{\log }_{x}}44.$
This implies that the values at the same positions can be equated and then the second part of the identity can be applied.
So, we will get $b=x,c=2$ and $a=44.$
So, by the second part of the identity, we can say that ${{b}^{c}}=a.$
So, we will get ${{x}^{2}}=44.$
Now, to get the value of the unknown, we will take the square root of the whole equation we have obtained.
We will get $x=\pm \sqrt{44}.$

Hence the solution is $x=\pm \sqrt{44}.$

Note: Sometimes, we may get the equation to be solved in terms of $\sqrt{x}.$ In that case, we may have to square the whole equation before or after some simplification so that we can find the solution in terms of $x$ itself.