What is the solution to this equation?
$9^x-1=2$
A. $\frac{1}{2}$
B. $-\frac{1}{2}$
C. 1
D. 2
Answer (1)
Add 1 to both sides of the equation: 9 x = 3 .
Rewrite 9 as 3 2 : ( 3 2 ) x = 3 .
Simplify the exponent: 3 2 x = 3 1 .
Equate the exponents and solve for x: x = 2 1 .
Explanation
Understanding the Problem We are given the equation 9 x − 1 = 2 and asked to find the value of x that satisfies this equation. Our goal is to isolate x and determine its value.
Isolating the Exponential Term First, we want to isolate the term with the exponent. To do this, we add 1 to both sides of the equation: 9 x − 1 + 1 = 2 + 1
9 x = 3
Rewriting the Base Now, we want to express both sides of the equation with the same base. We know that 9 = 3 2 , so we can rewrite the left side of the equation as ( 3 2 ) x . Thus, we have: ( 3 2 ) x = 3
Simplifying the Exponent Using the power of a power rule, which states that ( a m ) n = a mn , we can simplify the left side of the equation: 3 2 x = 3
Equating the Exponents Since the bases are now the same, we can equate the exponents. The right side of the equation can be written as 3 1 , so we have: 2 x = 1
Solving for x Finally, we solve for x by dividing both sides of the equation by 2: x = 2 1
Final Answer Therefore, the solution to the equation 9 x − 1 = 2 is x = 2 1 .
Examples
Exponential equations like this one are used in various real-world applications, such as modeling population growth, radioactive decay, and compound interest. For example, if you invest money in an account that compounds interest, the amount of money you have after a certain time can be modeled using an exponential equation. Understanding how to solve these equations allows you to predict future values and make informed decisions.
