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 8 as 2 3 .
Apply the power of a power rule: ( 2 3 ) x − 3 = 2 3 ( x − 3 ) .
Distribute the 3 in the exponent: 3 ( x − 3 ) = 3 x − 9 .
The equivalent equation is 2 4 x = 2 3 x − 9 .
Explanation
Problem Analysis 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 ) . This involves rewriting the right side of the equation using the fact that 8 is a power of 2 .
Rewriting the Base Since 8 = 2 3 , we can rewrite the right side of the equation as 8 x − 3 = ( 2 3 ) x − 3 .
Applying 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 .
Final Equivalent Equation Therefore, the equivalent equation is 2 4 x = 2 3 x − 9 .
Examples
Exponential equations are used to model various phenomena, such as population growth, radioactive decay, and compound interest. For example, if a bacteria population doubles every hour, the population size can be modeled by an exponential equation. Similarly, the decay of a radioactive substance can be modeled using an exponential equation, where the amount of substance remaining decreases exponentially over time. Understanding how to manipulate and solve exponential equations is crucial in these applications.
