To solve the exponential equation e^(x - 6) = 25, you need to take the natural logarithm of both sides of the equation. This is possible because the logarithm and the exponential functions are inverse operations. The exponent of e on the left side of the equation comes down as a coefficient when taking the natural logarithm.
1. Start with the given equation: e^(x - 6) = 25.
2. Take the natural logarithm on both sides of the equation: ln(e^(x - 6)) = ln(25)
This simplifies because ln and e are inverse functions, so ln(e^(x - 6)) simplifies to x - 6.
3. So now the equation is x - 6 = ln(25).
4. The next step is to isolate x, which is done by adding 6 to both sides of the equation: x - 6 + 6 = ln(25) + 6. This gives you the equation x = ln(25) + 6.
5. Solve this equation for x: x = ln(25) + 6.
When you calculate the natural logarithm of 25 and add 6, the result is roughly 9.218875824868201.
Therefore, the solution to the equation e^(x-6) = 25 is x = 9.218875824868201.