Select the correct answer.
What is the solution to this equation?
[tex]2\left(\frac{1}{49}\right)^{x-2}=14[/tex]
A. 3
B. [tex]\frac{3}{2}[/tex]
C. [tex]\frac{5}{2}[/tex]
D. [tex]-\frac{5}{2}[/tex]
Answer (2)
To find the solution for the equation 2 ( 49 1 ) x − 2 = 14 , we isolate the exponential part and rewrite it using a common base. After equating the exponents, we find that the solution is x = 2 3 , which corresponds to option B.
;
Divide both sides of the equation by 2: ( 49 1 ) x − 2 = 7 .
Rewrite 49 1 as 7 − 2 : ( 7 − 2 ) x − 2 = 7 .
Simplify the exponent: 7 − 2 ( x − 2 ) = 7 1 .
Equate the exponents and solve for x : − 2 ( x − 2 ) = 1 ⇒ x = 2 3 .
The solution to the equation is 2 3 .
Explanation
Problem Analysis We are given the equation 2\[\left(\frac{1}{49}\right)^{x-2}\]=14 and need to find the value of x that satisfies it.
Isolating the Exponential Term First, divide both sides of the equation by 2 to isolate the exponential term: ( 49 1 ) x − 2 = 2 14 = 7
Rewriting with a Common Base Rewrite 49 1 as 4 9 − 1 . Also, rewrite 49 as 7 2 , so 49 1 = ( 7 2 ) − 1 = 7 − 2 . The equation then becomes: ( 7 − 2 ) x − 2 = 7
Simplifying the Exponent Using the power of a power rule, simplify the left side: 7 − 2 ( x − 2 ) = 7 1
Equating Exponents Since the bases are equal, we can equate the exponents: − 2 ( x − 2 ) = 1
Solving for x Solve for x :
− 2 x + 4 = 1 − 2 x = 1 − 4 − 2 x = − 3 x = − 2 − 3 = 2 3
Final Answer Therefore, the solution to the equation is x = 2 3 .
Examples
Exponential equations are useful in modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in an account with compound interest, the amount of money you have after a certain time can be modeled using an exponential equation. Solving such equations helps you determine how long it will take for your investment to reach a certain value.
