Question

Solve the following equation: \[\mathop {\log }\nolimits_{x - 1}^3 = 2\]

Answer 1

Hint: Use the basic definition of logarithmic function to reverse the given equation. Recall how we take \[\log \]in any given equation and use the same property to get the equation back in the form of the quadratic equation and then solve the obtained equation.

Complete step-by-step solution:
We are given with the equation\[\mathop {\log }\nolimits_{x - 1}^3 = 2\]then we know that,
\[ \mathop {\log }\nolimits_a^c = b \\
   \Rightarrow {a^b} = c \]
Here we have \[a = (x - 1),{\kern 1pt} {\kern 1pt} b = 2,{\kern 1pt} {\kern 1pt} c = 3\]
Then using this definition we write our given equation as,
\[
  {(x - 1)^2} = 3 \\
   \Rightarrow {x^2} - 2x + 1 = 3 \\
   \Rightarrow {x^2} - 2x - 2 = 0 \]
Now, using quadratic formula to find the roots of the above obtained equation i.e.
\[
  x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} \\
   \Rightarrow x = \dfrac{{2 \pm \sqrt {{2^2} + 4.1.2} }}{{2.1}} \\
   \Rightarrow x = \dfrac{{2 \pm \sqrt {4 + 8} }}{2} \\
   \Rightarrow x = \dfrac{{2 \pm \sqrt {12} }}{2} \\
   \Rightarrow x = \dfrac{{2 \pm \sqrt {4.3} }}{2} \\
   \Rightarrow x = \dfrac{{2 \pm 2\sqrt 3 }}{2} \\
   \Rightarrow x = \dfrac{{1 \pm \sqrt 3 }}{1} \\
   \Rightarrow x = 1 + \sqrt 3 ,{\kern 1pt} {\kern 1pt} 1 - \sqrt 3 \]
Hence the roots of the above equation are \[x = 1 + \sqrt 3 ,{\kern 1pt} {\kern 1pt} 1 - \sqrt 3 \]
Additional information: The first approach to convert the logarithmic equation into a quadratic equation is very important. A quadratic is a sort of example that offers a variable extended via itself — an operation called squaring. This language derives from the vicinity of a square being its side length increased by way of itself. The word "quadratic" comes from quadratum, the Latin word for square. Quadratic equations are virtually used in everyday life, as whilst calculating regions, figuring out a product's earnings or formulating the velocity of an item. Quadratic equations consult with equations with at least one squared variable, with the most standard form being. Quadratic equations are used to represent different functions, different real life processes and used in various fields like chemistry, biology, physics and many other fields to solve and get the desired solution.

Note: The basic idea of converting the logarithmic function into a quadratic equation is important. It is also important that we know various ways, techniques, shortcuts and methods to solve different types of quadratic equations. Middle term factoring, Sridharacharya method or Quadratic formula, graphical methods are some techniques of solving the roots of a quadratic equation.