Sure, let's solve for x in the equation e^(x+2) = 4, using principles from algebra and properties of logarithms.
Step 1: Taking natural logarithms on both sides
Since both sides of the equation are equal, we can take the logarithm, specifically the natural logarithm denoted as ln, of both sides without changing the solution.
This gives us the equation ln(e^(x+2)) = ln(4).
Step 2: Simplifying the left-hand side
We can simplify the left-hand side of the equation thanks to the property of logarithms that states ln(a^b) = b * ln(a). Given that the natural logarithm of e, ln(e), is 1, this further simplifies the left-hand side of the equation:
We get x+2 = ln(4).
Step 3: Isolating x
To solve for x, we subtract 2 from both sides of the equation to isolate x:
x = ln(4) - 2
Therefore, the solution to the equation e^(x+2)=4, expressed in terms of natural logarithms is x = ln(4) - 2.
To obtain a numerical solution, we can substitute ln(4) with its approximate decimal value. From the natural logarithm table, we find that ln(4) is about 1.386294. Subtracting 2 from 1.386294 gives us -0.6137056388801094.
Therefore, the numerical solution to the problem is roughly x = -0.6137056388801094.
In other words, e raised to the power of this value plus 2 (x+2) equals 4.