Question

How do you solve ${\log _4}({x^2} - 9) - {\log _4}(x + 3) = 3$?

Answer 1

Hint: Here we have to solve the logarithmic function. Logarithmic functions are inverse of exponential functions. In order to solve this logarithmic function we will use logarithmic quotient rule which is given by the formula ${\log _b}\left( {\dfrac{P}{Q}} \right) = {\log _b}P - {\log _b}Q$ and algebraic identity of difference of two squares i.e., ${a^2} - {b^2} = (a - b)(a + b)$.

Complete step by step answer:
Here we have to solve the logarithmic function ${\log _4}({x^2} - 9) - {\log _4}(x + 3) = 3$.Logarithmic functions are inverse of exponential function and we can express any exponential function in logarithmic form. A logarithm is used to raise the power of a number to get a certain number.We will use logarithms quotient rule which states that the logarithm of the ratio of two numbers is equal to the difference between the logarithm of the numerator and denominator which is given by the formula.
${\log _b}\left( {\dfrac{P}{Q}} \right) = {\log _b}P - {\log _b}Q$
We have ${\log _4}({x^2} - 9) - {\log _4}(x + 3) = 3$

We know that ${a^2} - {b^2} = (a - b)(a + b)$
So, we can write $({x^2} - 9) = ({x^2} - {3^2})$
By applying the above formula in the expression. We get,
$({x^2} - {3^2}) = (x - 3)(x + 3)$
Substituting ${x^2} - 9$ by $(x - 3)(x + 3)$ in the equation. We get,
\[ \Rightarrow {\log _4}(x - 3)(x + 3) - {\log _4}(x + 3)\]$ = 3$
We know that ${\log _b}P - {\log _b}Q = {\log _b}\left( {\dfrac{P}{Q}} \right)$.
Applying this formula in the function. We get,
$ \Rightarrow $${\log _4}\left( {\dfrac{{(x - 3)(x + 3)}}{{x + 3}}} \right) = 3$

Cancelling out $(x + 3)$ term in the above equation. We get,
$ \Rightarrow {\log _4}(x - 3) = 3$
As logarithmic function is the inverse of exponential function we can write the above equation as
$ \Rightarrow x - 3 = {4^3}$
Simplifying the cube root of $4$. We get,
$ \Rightarrow x - 3 = 64$
Shifting $3$ to the right side of the equation. We get,
$ \Rightarrow x = 67$

Hence $x = 67$.

Note: Logarithm can be defined as an exponent or power to which a base must be raised to obtain a given number. They can be expressed as, $m$ is the logarithm of $n$ to the base $b$ if ${b^m} = n$, so, it can be written as $m = {\log _b}n$. The commonly used logarithms are the logarithm to the base $10$ which is also known as decimal logarithm because of its base and the value of ${\log _{10}}10 = 1$.