Solving for exponents is the process of finding the power (exponent) to which a base number must be raised to obtain a given result. This is useful in various mathematical and real-world applications, including compound interest, population growth, and scientific calculations.

To solve for exponents, we use logarithms. Logarithms allow us to convert exponential equations into linear equations, making it easier to solve for the unknown exponent.

The formula to solve for exponents is:

\[ n = \frac{\log(y)}{\log(x)} \]

Where:

- n is the exponent we're solving for
- x is the base
- y is the result

- Start with the equation: \(x^n = y\)
- Take the logarithm of both sides: \(\log(x^n) = \log(y)\)
- Use the logarithm rule \(\log(a^b) = b \cdot \log(a)\): \(n \cdot \log(x) = \log(y)\)
- Divide both sides by \(\log(x)\): \(n = \frac{\log(y)}{\log(x)}\)
- Calculate the result using the values of x and y

Let's solve for n in the equation \(2^n = 32\):

- We have x = 2 and y = 32
- Apply the formula: \(n = \frac{\log(32)}{\log(2)}\)
- Calculate: n ≈ 5
- Verify: \(2^5 = 32\)

Therefore, the exponent n is 5.

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