For what value of [tex]$x$[/tex] does [tex]$3^{2 x}=9^{3 x-4} ?$[/tex]
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Answer (1)
Rewrite the equation with a common base: 3 2 x = ( 3 2 ) 3 x − 4 .
Simplify the equation using the power of a power rule: 3 2 x = 3 6 x − 8 .
Set the exponents equal to each other: 2 x = 6 x − 8 .
Solve the linear equation for x : x = 2 .
Explanation
Understanding the Problem We are given the equation 3 2 x = 9 3 x − 4 and we want to find the value of x that satisfies this equation. Notice that the bases of the exponential terms are 3 and 9, where 9 is a power of 3 ( 9 = 3 2 ).
Rewriting with a Common Base To solve this equation, we need to rewrite it with a common base. Since 9 = 3 2 , we can rewrite the equation as 3 2 x = ( 3 2 ) 3 x − 4 .
Simplifying the Equation Now, we simplify the right side of the equation using the power of a power rule: ( a m ) n = a mn . This gives us 3 2 x = 3 2 ( 3 x − 4 ) .
Equating the Exponents Since the bases are equal, we can set the exponents equal to each other: 2 x = 2 ( 3 x − 4 ) .
Solving for x Now, we solve the linear equation for x : 2 x = 6 x − 8 .
Isolating x Isolate x by subtracting 6 x from both sides: 2 x − 6 x = − 8 , which simplifies to − 4 x = − 8 .
Finding the Value of x Finally, divide both sides by − 4 to find the value of x : x = − 4 − 8 = 2 . Therefore, the value of x that satisfies the equation is 2.
Checking the Answer We can check our answer by plugging x = 2 back into the original equation: 3 2 ( 2 ) = 3 4 = 81 and 9 3 ( 2 ) − 4 = 9 6 − 4 = 9 2 = 81 . Since both sides are equal, our answer is correct.
Examples
Exponential equations are useful in modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population doubles every hour, we can use an exponential equation to predict the population size after a certain number of hours. Similarly, in finance, compound interest calculations rely on exponential functions to determine the future value of an investment.
