Question

How do you solve $\ln x + \ln (x + 1) = \ln 12$ ?

Answer 1

Hint : In order to determine the value of the above question, rewrite the expression using the property of logarithm ${\log _b}(m) + {\log _b}(n) = {\log _b}(mn)$ and take antilogarithm on both side to remove logarithm from the expression then use the splitting up the middle method to find the solution of the quadratic equation formed.
Formula:
${\log _b}(m) + {\log _b}(n) = {\log _b}(mn)$

Complete step-by-step answer :
We are Given an expression $\ln x + \ln (x + 1) = \ln 12$
Now, rewriting the expression using the property of logarithm ${\log _b}(m) + {\log _b}(n) = {\log _b}(mn)$
$\ln \left( {x(x + 1)} \right) = \ln 12$
Taking antilogarithm on both sides ,this will remove the logarithm from both the sides, our expression now becomes

$$\Rightarrow x(x + 1) = 12 \\\ \Rightarrow {x^2} + x = 12 \\\ \Rightarrow {x^2} + x - 12 = 0 \;$$

Expression has become a quadratic equation, and to solve this we’ll use splitting up the middle term method.
Follow below steps to split the middle term
Step 1: calculate the product of coefficient of ${x^2}$ and the constant term which comes to be
$= - 12 \times 1 = - 12$
Step 2:find the 2 factors of the number -12 such that the weather addition or subtraction of those numbers is equal to the middle term or coefficient of x and the product of those factors results in the value of constant .
So if we factorize 12, the answer comes to be 4and 3 as $4 - 3 = 1$ that is the middle term . and $4 \times 3 = 12$ which is perfectly equal to the constant value.
Now writing the middle term sum of the factors obtained, so equation becomes

$$\Rightarrow {x^2} + 4x - 3x - 12 = 0 \\\ \Rightarrow x(x + 4) - 3(x + 4) = 0 \\\ \Rightarrow (x + 4)(x - 3) = 0 \;$$

$x + 4 = 0 \\\ \Rightarrow x = - 4 \\\ x - 3 = 0 \\\ \Rightarrow x = 3 \;$
Value of x can be $- 4,3$
Since $\ln x$ is not defined for the negative values of x so $x = 3$
Therefore, the value of $x = 3$ .
So, the correct answer is “ $x = 3$ ”.

Note : 1.Value of constant ‘e’ is equal to $2.71828$ .
2.A logarithm is basically the reverse of a power or we can say when we calculate a logarithm of any number, we actually undo an exponentiation.
3.Any multiplication inside the logarithm can be transformed into addition of two separate logarithm values.
${\log _b}(mn) = {\log _b}(m) + {\log _b}(n)$
4. Any division inside the logarithm can be transformed into subtraction of two separate logarithm values.
${\log _b}\left( {\dfrac{m}{n}} \right) = {\log _b}(m) - {\log _b}(n)$
5. Any exponent value on anything inside the logarithm can be transformed and moved out of the logarithm as a multiplier and vice versa.
$n\log m = \log {m^n}$
6.Quadratic Equation: A quadratic equation is a equation which can be represented in the form of $a{x^2} + bx + c$ where $x$ is the unknown variable and a,b,c are the numbers known where $a \ne 0$ .If $a = 0$ then the equation will become linear equation and will no more quadratic .
The degree of the quadratic equation is of the order 2.