Question

How do you solve $ {7^{2x - 1}} = 343 $ ?

Answer 1

Hint: In order to determine the solution of the given exponential function, write the base of both right-hand sides in the form of number 7 raised to power some $ n $ . Since the bases of both sides of the equation are the same then use the property of the 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 answer:
We are given exponential equation having variable $ x $ i.e. $ {7^{2x - 1}} = 343 $ .
In order to solve this equation, we will be using rules and properties of exponent to simplify the equation and later find the value of variable $ x $
 $ {7^{2x - 1}} = 343 $
Try to write the base part right-hand side of the equation in the form of number 7 raised to power some n.
So, 343 can be written as $ 343 = 7 \times 7 \times 7 = $ $ {7^3} $ . Rewriting the original equation by replacing 343 with $ {7^3} $ . Our equation now becomes
 $ {7^{2x - 1}} = {7^3} $
Since, on both sides of the equation the base is equal so by using the rule of exponent which states that $ {a^m} = {a^n} \to m = n $ , we get
 $ \Rightarrow 2x - 1 = 3 $
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 2x = 3 + 1 $
Combining like term
 $ \Rightarrow 2x = 4 $
Dividing both sides of the equation with coefficient of $ x $ i.e. 2 , we get
 $
   \Rightarrow \dfrac{{2x}}{2} = \dfrac{4}{2} \\
   \Rightarrow x = 2 \;
  $
Therefore, the solution of the exponential function $ {7^{2x - 1}} = 343 $ is $ x = 2 $ .
So, the correct answer is “x=2”.

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.