Question

How do you solve the exponential equation $ {8^{5x}} = {16^{3x + 4}} $ ?

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 2 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. $ {8^{5x}} = {16^{3x + 4}} $ .
In order to solve this equation, we will using rules and properties of exponent to simplify the equation and later find the value of variable $ x $
 $ {8^{5x}} = {16^{3x + 4}} $
Try to write the base part of both the sides of the equation in the form of number 2 raised to power some n.
So 8 can be written as $ {2^3} $ and similarly 16 can also be written as $ {2^4} $ . Rewriting the original equation by replacing 8 with $ {2^3} $ and 16 with $ {2^4} $ . Our equation now becomes
 $
   \Rightarrow {8^{5x}} = {16^{3x + 4}} \\
   \Rightarrow {\left( {{2^3}} \right)^{5x}} = {\left( {{2^4}} \right)^{3x + 4}} \\
  $
Now using the most important property of exponent on both the terms 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 {\left( 2 \right)^{15x}} = {\left( 2 \right)^{4\left( {3x + 4} \right)}} \\
   \Rightarrow {\left( 2 \right)^{15x}} = {\left( 2 \right)^{12x + 16}} \;
  $
Since, on both sides of the equation the base are equal so by the rule of exponent $ {a^m} = {a^n} \to m = n $ , we get
 $ \Rightarrow 15x = 12x + 16 $
We have obtained a linear equation, so transposing the terms having variable $ x $ from right-hand side to left-hand side using the rules of transposing of terms, we obtain
 $ \Rightarrow 15x - 12x = 16 $
Combining like term
 $ \Rightarrow 3x = 16 $
Now dividing both sides of the equation with the coefficient of the variable $ x $ i.e.3
 $
   \Rightarrow \dfrac{{3x}}{3} = \dfrac{{16}}{3} \\
   \Rightarrow x = \dfrac{{16}}{3} \;
  $
Therefore the solution of the exponential function $ {8^{5x}} = {16^{3x + 4}} $ is $ x = \dfrac{{16}}{3} $ .
So, the correct answer is “$ x = \dfrac{{16}}{3} $”.

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 .