Question

How do you solve the exponential equation $ {3^{x - 7}} = {27^{2x}} $ ?

Answer 1

Hint: In order to determine the solution of the given exponential function, write the base of both sides in the form of number 3 raised to power some $ n $ . Rewrite the equation using the rule of exponent $ {\left( {{a^m}} \right)^n} = {a^{m \times n}} $ .Since the base of both sides of equation are same then use the property of exponent that $ {a^m} = {a^n} \to m = n $ to obtain a linear equation. Solve the linear equation obtained to get the required result.

Complete step by step solution:
We are given an exponential function having variable $ x $ i.e. $ {3^{x - 7}} = {27^{2x}} $ .
In order to solve this equation, we will be using rules and properties of exponent to simply the equation and later find the value of variable $ x $
 $ {3^{x - 7}} = {27^{2x}} $
Try to write the base part of both the sides of the equation in the form of number 3 raised to power some n.
So, 27 can be written as $ {3^3} $ . Rewriting the original equation by replacing 27 with $ {3^3} $ . Our equation now becomes
 $ {3^{x - 7}} = {\left( {{3^3}} \right)^{2x}} $
Now using the most important property of exponent on the RHS that when a number having some exponent raised to some exponent value then both the exponent values get multiplied as $ {\left( {{a^m}} \right)^n} = {a^{m \times n}} $ . We get
 $
   \Rightarrow {3^{x - 7}} = {\left( 3 \right)^{3\left( {2x} \right)}} \\
   \Rightarrow {3^{x - 7}} = {3^{6x}} \;
  $
Since, on both sides of the equation the base is equal so by the rule of exponent $ {a^m} = {a^n} \to m = n $ , we get
 $ \Rightarrow x - 7 = 6x $
We have obtained a linear equation, so transposing the terms having variable $ x $ from right-hand side to left-hand side and constant terms from LHS to RHS using the rules of transposing of terms, we obtain
 $ \Rightarrow x - 6x = 7 $
Combining like term
 $ \Rightarrow - 5x = 7 $
Now dividing both sides of the equation with the coefficient of the variable $ x $ i.e.-5
 $
   \Rightarrow \dfrac{{ - 5x}}{{ - 5}} = \dfrac{7}{{ - 5}} \\
   \Rightarrow x = - \dfrac{7}{5} \;
  $
Therefore, the solution of the exponential function $ {3^{x - 7}} = {27^{2x}} $ is $ x = - \dfrac{7}{5} $
So, the correct answer is $ x = - \dfrac{7}{5} $.

Note: 1. Value of the exponential constant ‘e’ is equal to 2.7182818.
2.Use the properties and rules of exponent carefully while solving the equations.
3. We have obtained a single value for variable $ x $ as linear equations always have only one solution.
4. Don’t forget to cross-check your answer at the end.