For what value of $x$ does $3^{4 x}=27^{x-3}$?
Answer (2)
Rewrite the equation using the fact that 27 = 3 3 , which gives 3 4 x = ( 3 3 ) x − 3 .
Simplify the right side using the power of a power rule: 3 4 x = 3 3 ( x − 3 ) .
Set the exponents equal to each other: 4 x = 3 ( x − 3 ) .
Solve the resulting linear equation for x : x = − 9 .
The solution is − 9 .
Explanation
Understanding the Problem We are given the equation 3 4 x = 2 7 x − 3 and need to find the value of x that satisfies it.
Rewriting the Equation We can rewrite the equation using the fact that 27 = 3 3 . Substituting this into the original equation, we get 3 4 x = ( 3 3 ) x − 3 .
Simplifying the Exponents Using the power of a power rule, ( a m ) n = a mn , we can simplify the right side of the equation: 3 4 x = 3 3 ( x − 3 ) .
Equating the Exponents Since the bases are equal, the exponents must be equal. Therefore, we set the exponents equal to each other: 4 x = 3 ( x − 3 ) .
Solving for x Now, we solve the resulting linear equation for x :
4 x = 3 ( x − 3 )
4 x = 3 x − 9
4 x − 3 x = − 9
x = − 9
Final Answer Therefore, the value of x that satisfies the equation is − 9 .
Examples
Exponential equations are useful in modeling 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 size at any given time. Similarly, in finance, compound interest calculations rely on exponential growth to determine the future value of an investment.
To solve the equation 3 4 x = 2 7 x − 3 , we rewrite it using the fact that 27 = 3 3 . After simplification, we set the exponents equal and solve, resulting in x = − 9 .
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