Which equation is equivalent to $2^{4 x}=8^{x-3}$ ?
A. $2^{4 x}=2^{2 x-3}$
B. $2^{4 x}=2^{2 x-6}$
C. $2^{4 x}=2^{3 x-3}$
D. $2^{4 x}=2^{3 x-9}$
Answer (1)
Rewrite the base 8 as 2 3 .
Apply the power of a power rule: ( a m ) n = a mn .
Simplify the exponent.
The equivalent equation is 2 4 x = 2 3 x − 9 .
Explanation
Understanding the Problem We are given the equation 2 4 x = 8 x − 3 and asked to find an equivalent equation in the form 2 4 x = 2 f ( x ) . The key idea here is to express both sides of the equation with the same base, which is 2 in this case.
Rewriting with the Same Base Since 8 = 2 3 , we can rewrite the right side of the equation as 8 x − 3 = ( 2 3 ) x − 3 .
Applying the Power of a Power Rule Using the power of a power rule, which states that ( a m ) n = a mn , we have ( 2 3 ) x − 3 = 2 3 ( x − 3 ) .
Simplifying the Exponent Now, we simplify the exponent by distributing the 3: 3 ( x − 3 ) = 3 x − 9 . Therefore, 8 x − 3 = 2 3 x − 9 .
Finding the Equivalent Equation So, the equivalent equation is 2 4 x = 2 3 x − 9 .
Final Answer Therefore, the equation equivalent to 2 4 x = 8 x − 3 is 2 4 x = 2 3 x − 9 .
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, in finance, compound interest is calculated using exponential functions. If you invest P dollars at an annual interest rate r compounded n times per year, the amount A you'll have after t years is given by A = P ( 1 + n r ) n t . Understanding how to manipulate exponential equations allows you to solve for any of the variables, such as the time it takes to reach a certain investment goal.
