Question

How do you solve $2 - {\log _3}\sqrt {{x^2} + 17} = 0$?

Answer 1

Hint: According to the question we have to solve the given logarithmic expression which is $2 - {\log _3}\sqrt {{x^2} + 17} = 0$as given in the question. So, to solve the given expression first of all we have to rearrange the terms of the logarithmic expression for that we have to take all the log terms in the right hand side of the expression.
Now, to simplify the log terms we have to use the formula which is as mentioned below:

Formula used:
$
 x = {\log _a}y \\
   \Rightarrow {(a)^x} = y.................(A) \\
 $
Now, we have to apply the square in the both left and right sides of the expression as obtained after applying the formula (A).
Now, to solve the obtained expression in the form of a quadratic expression we just have to take all the integers one side and all the variables to the other sides of the expression.
Now, we just have to determine the square root of the expression obtained with the help of the formula as mentioned below:

Formula used:
$ \Rightarrow \sqrt {a \times a} = \pm a.....................(B)$

Complete step by step solution:
Step 1: First of all we have to rearrange the terms of the logarithmic expression for that we have to take all the log terms in the right hand side of the expression. Hence,
$ \Rightarrow 2 = {\log _3}\sqrt {{x^2} + 17} $
Step 2: Now, to simplify the log terms we have to use the formula (A) which is as mentioned in the solution hint. Hence,
$
   \Rightarrow {(3)^2} = \sqrt {{x^2} + 17} \\
   \Rightarrow 9 = \sqrt {{x^2} + 17} \\
 $
Step 3: Now, we have to apply the square in the both left and right sides of the expression as obtained after applying the formula (A). Hence,
$
   \Rightarrow {9^2} = {(\sqrt {{x^2} + 17} )^2} \\
   \Rightarrow 81 = {x^2} + 17 \\
 $
Step 4: Now, to solve the obtained expression in the form of a quadratic expression we just have to take all the integers one side and all the variables to the other sides of the expression.
$
   \Rightarrow {x^2} = 81 - 17 \\
   \Rightarrow {x^2} = 64 \\
 $
Step 5: Now, we just have to determine the square root of the expression obtained with the help of the formula (B) as mentioned in the solution hint. Hence,
$
   \Rightarrow x = \sqrt {64} \\
   \Rightarrow x = \sqrt {8 \times 8} \\
   \Rightarrow x = \pm 8 \\
 $

Final solution: Hence, with the help of the formulas (A) and (B) we have determined the solution of $2 - {\log _3}\sqrt {{x^2} + 17} = 0$ which is $x = \pm 8$.

Note:
To determine the value of the given logarithmic expression it is necessary that we have to eliminate the given long term with the help of the formula (A) as mentioned in the solution hint.
A number is in a form of square root such as $\sqrt a $ and if we can break the number in the form of $\sqrt {b \times b} $ then we can easily determine the square root of that given number.