Solve $(\sqrt{7})^{6 x}=49^{x-6}$
Answer (1)
Rewrite the equation using the property that 49 = 7 2 and 7 = 7 2 1 .
Simplify the exponents using the power of a power rule: 7 3 x = 7 2 x − 12 .
Equate the exponents: 3 x = 2 x − 12 .
Solve for x : x = − 12 .
Explanation
Problem Analysis We are given the equation ( 7 ) 6 x = 4 9 x − 6 . Our goal is to solve for x .
Rewriting the Equation First, we rewrite the equation using the property that 49 = 7 2 and 7 = 7 2 1 . This gives us ( 7 2 1 ) 6 x = ( 7 2 ) x − 6 .
Simplifying Exponents Next, we simplify the exponents. Using the power of a power rule, we have 7 3 x = 7 2 ( x − 6 ) , which simplifies to 7 3 x = 7 2 x − 12 .
Equating Exponents Since the bases are equal, we can equate the exponents: 3 x = 2 x − 12 .
Solving for x Now, we solve for x . Subtracting 2 x from both sides gives 3 x − 2 x = − 12 , which simplifies to x = − 12 .
Final Answer Therefore, the solution to the equation ( 7 ) 6 x = 4 9 x − 6 is x = − 12 .
Examples
Exponential equations are useful in modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population doubles every 10 years, we can use an exponential equation to predict the population at any given time. Similarly, in finance, compound interest calculations rely on exponential growth to determine the future value of an investment.
