What equation is equivalent to $9^{x-3}=729$?
A. $9^{x-3}=9^{81}$
B. $9^{x-3}=9^3$
C. $3^{x-3}=3^6$
D. $3^{2 x-3}=3^6$
Answer (2)
Express 729 as a power of 9: 729 = 9 3 .
The equation 9 x − 3 = 729 is equivalent to 9 x − 3 = 9 3 .
Express 9 and 729 as powers of 3: 9 = 3 2 and 729 = 3 6 .
The equation 9 x − 3 = 729 can be written as ( 3 2 ) x − 3 = 3 6 , which simplifies to 3 2 ( x − 3 ) = 3 6 or 3 2 x − 6 = 3 6 .
The equivalent equation is 9 x − 3 = 9 3 .
Explanation
Understanding the Problem We are given the equation 9 x − 3 = 729 and asked to find an equivalent equation from the given options.
Expressing 729 as a power of 9 First, let's express 729 as a power of 9. We know that 9 3 = 729 . Therefore, the equation 9 x − 3 = 729 is equivalent to 9 x − 3 = 9 3 .
Expressing 9 and 729 as powers of 3 Now, let's express 9 and 729 as powers of 3. We have 9 = 3 2 and 729 = 3 6 . So, the equation 9 x − 3 = 729 can be written as ( 3 2 ) x − 3 = 3 6 , which simplifies to 3 2 ( x − 3 ) = 3 6 or 3 2 x − 6 = 3 6 .
Comparing with the given options Comparing the derived equations with the given options, we see that 9 x − 3 = 9 3 is one of the options. Also, 3 2 x − 6 = 3 6 is equivalent to 3 2 x − 3 = 3 6 only if we add -3 on the left side instead of -6 which is incorrect.
Final Answer Therefore, the equivalent equation is 9 x − 3 = 9 3 .
Examples
Exponential equations are used in various fields, such as calculating 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 after a certain number of years. Similarly, in finance, compound interest calculations rely on exponential growth to determine the future value of an investment.
The equation 9 x − 3 = 729 is equivalent to 9 x − 3 = 9 3 . This is because both expressions represent the same value of 729. Thus, the correct choice is Option B.
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