What is the solution to this equation?
$9^x - 1 = 2$
The given equation is \(9^{\alpha} - 1 = 2\). Our objective is to determine the value of \(\alpha\) that satisfies this equation. This is an exponential equation where the unknown \(\alpha\) is located in the exponent. To solve it, we need to isolate the term containing \(\alpha\) and then express both sides of the equation in terms of a common base.
To isolate the term \(9^{\alpha}\), we add \(1\) to both sides of the equation:
\[9^{\alpha} - 1 + 1 = 2 + 1\]
\[9^{\alpha} = 3\]
Now the equation is in a simpler form, where we have an exponential expression set equal to a constant.
Notice that both \(9\) and \(3\) can be expressed as powers of the base \(3\). Specifically, \(9 = 3^2\). Substituting this into our equation gives:
\[(3^2)^{\alpha} = 3^1\]
Using the power of a power property for exponents, \((a^m)^n = a^{m \times n}\), we can rewrite the left side:
\[3^{2\alpha} = 3^1\]
Since the bases are the same, we can equate the exponents:
\[2\alpha = 1\]
Dividing both sides by \(2\), we find:
\[\alpha = \frac{1}{2}\]
Comparing our result, \(\alpha = \frac{1}{2}\), with the provided options, we can conclude that the correct choice is the final option. Therefore, the solution to the equation \(9^{\alpha} - 1 = 2\) is \(\alpha = \frac{1}{2}\). The final answer is \(\boxed{\frac{1}{2}}\).
Exponential growth is a fundamental concept in fields ranging from finance to biology. For example, if an investment's value grows such that its multiplier follows a pattern similar to \(9^{\alpha}\), calculating the exact time \(\alpha\) required to reach a certain threshold involves solving exponential equations like this one. Understanding how to solve for unknown exponents helps in predicting growth, planning investments, and modeling real-world processes.