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.