Finding the variable in an exponent involves using logarithms to "undo" the exponentiation. Here's how you can do it, based on the provided reference:
Steps to Solve for a Variable in an Exponent
Here’s a breakdown of the process:
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Take the log of both sides: Apply a logarithm (usually base 10 or the natural logarithm, ln) to both sides of the equation. This is the crucial first step in isolating the variable.
- Example: If you have the equation 2x = 8, taking the log of both sides gives you log(2x) = log(8).
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Use the power rule to drop down the exponent: The power rule of logarithms states that logb(ac) = c * logb(a). Apply this rule to move the variable exponent down as a coefficient.
- Example (continuing from above): log(2x) = log(8) becomes x * log(2) = log(8).
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Divide both sides by the appropriate log to isolate the variable: Divide both sides of the equation by the logarithmic term that is now multiplying the variable. This isolates the variable x.
- Example (continuing from above): To solve x * log(2) = log(8), divide both sides by log(2): x = log(8) / log(2).
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Solve for the variable: Calculate the value of the logarithms and perform the division to find the value of the variable.
- Example (continuing from above): x = log(8) / log(2) = 0.903 / 0.301 ≈ 3. Therefore, x = 3.
Example Summary Table
| Step | Action | Example |
|---|---|---|
| 1. Take the log of both sides | Apply log to both sides of the equation | 2x = 8 -> log(2x) = log(8) |
| 2. Power Rule | Move exponent down as a coefficient | log(2x) = log(8) -> x * log(2) = log(8) |
| 3. Isolate the Variable | Divide by the log term | x * log(2) = log(8) -> x = log(8) / log(2) |
| 4. Solve | Calculate the value | x = log(8) / log(2) ≈ 3 |
By following these steps, you can successfully find the variable of an exponent. Remember to use the properties of logarithms correctly!
