Question

How do you solve ${\log _4}\left( {{x^2} - 4} \right) - {\log _4}\left( {x + 2} \right) = 2$ ?

Answer 1

Hint : To solve the problem we should know about the following term.
Logarithm: The exponent that indicates the power to which a base number is raised to produce a given number. the logarithm of $100$ to the base $10$ is $2$

Complete step by step solution:
As given in the quest
${\log _4}\left( {{x^2} - 4} \right) - {\log _4}\left( {x + 2} \right) = 2$
Let’s determine the set of real numbers where this equation makes sense. It’s necessary because we might engage in non- invariant transformation of this equation, which might produce extra solutions or we might lose certain solutions in the course of transformation. We will use only invariant transformations. It’s still a good practice to determine what kind of solutions our equation allows.
Any logarithm with a base $4$ (any other positive base) is defined only for positive arguments. That why we have restrictions:
${x^2} - 4 > 0$
$x + 2 > 0$
The first condition is equivalent to ${x^2} > 4$ or $\left| x \right| > 2$ a combination of two inequalities:
$x < \- 2$ or,
$x > 2$
The second condition is equivalent to
$x > - 2$
The only condition that satisfies condition (3) and either (1) or (2) above is
$x > 2$
Now let’s examine the problem at hand.
Recall that
${\log _c}(a \times b) = {\log _c}(a) + {\log _c}(b)$
Applying this to our equation and using an identify ${x^2} - 4 = (x - 2) \times (x + 2)$ , we can rewrite it as $\log$
${\log _4}\left( {x - 2} \right) + {\log _4}\left( {x + 2} \right) - {\log _4}(x + 2) = 2$
Or,
$\Rightarrow {\log _4}(x - 2) = 2$
$\Rightarrow {4^2} = x - 2$ or,
$\Rightarrow 16 = x - 2$ or,
$\Rightarrow x = 18$
Hence, $x = 18$
So, the correct answer is “ $x = 18$ ”.

Note : More generally, exponentiation allow any positive real number as base to be raised to any real power, always producing a positive result, so ${\log _b}\left( x \right)$ for any two positive real numbers $b$ and $x$ where $b$ is not equal to $1$ , is always a unique real number $y$ .