Which equation is equivalent to $\left(\frac{1}{3}\right)^x=27^{x+2}$ ?
A. $3^x=3^{-3 x+2}$
B. $3^x=3^{3 x+6}$
C. $3^{-x}=3^{3 x+2}$
D. $-x \quad 3 x+6$
Answer (1)
Rewrite the left side of the equation: ( 3 1 ) x = 3 − x .
Rewrite the right side of the equation: 2 7 x + 2 = ( 3 3 ) x + 2 .
Simplify the right side: ( 3 3 ) x + 2 = 3 3 ( x + 2 ) = 3 3 x + 6 .
The equivalent equation is: 3 − x = 3 3 x + 6 .
Explanation
Understanding the Problem We are given the equation ( 3 1 ) x = 2 7 x + 2 and we want to find an equivalent equation in the form 3 a = 3 b .
Rewriting the Left Side First, we rewrite the left side of the equation. Since 3 1 = 3 − 1 , we have ( 3 1 ) x = ( 3 − 1 ) x = 3 − x .
Rewriting the Right Side Next, we rewrite the right side of the equation. We know that 27 = 3 3 , so we have 2 7 x + 2 = ( 3 3 ) x + 2 .
Simplifying the Right Side Now, we simplify the right side using the power of a power rule: ( 3 3 ) x + 2 = 3 3 ( x + 2 ) = 3 3 x + 6 .
Finding the Equivalent Equation Therefore, the equivalent equation is 3 − x = 3 3 x + 6 .
Final Answer The equivalent equation to ( 3 1 ) x = 2 7 x + 2 is 3 − x = 3 3 x + 6 .
Examples
Understanding exponential equations is crucial in various fields, such as calculating compound interest. For instance, if you invest money in an account with continuously compounded interest, the amount you'll have after a certain time can be modeled using an exponential equation. By manipulating and solving these equations, you can determine how long it will take for your investment to reach a specific goal, or what interest rate you need to achieve that goal. This skill is also applicable in understanding population growth, radioactive decay, and many other real-world phenomena.
