To solve the logarithmic equation [tex]\(\ln(x) = 6\)[/tex] for [tex]\(x\)[/tex], we can follow these steps:
1. Understand the Logarithmic Equation:
The natural logarithm function, [tex]\(\ln(x)\)[/tex], gives us the power to which [tex]\(e\)[/tex] (the base of the natural logarithm, approximately equal to 2.71828) must be raised to result in [tex]\(x\)[/tex]. So, [tex]\(\ln(x) = 6\)[/tex] means we are looking for a number [tex]\(x\)[/tex] such that [tex]\(e\)[/tex] raised to which power equals [tex]\(x\)[/tex].
2. Exponentiate Both Sides:
To get rid of the natural logarithm, exponentiate both sides of the equation with base [tex]\(e\)[/tex]:
[tex]\[
e^{\ln(x)} = e^6
\][/tex]
3. Simplify:
Since [tex]\(e^{\ln(x)} = x\)[/tex], we simplify the equation to:
[tex]\[
x = e^6
\][/tex]
4. Calculate the Value:
Using the properties of exponentiation and the value of [tex]\(e \approx 2.71828\)[/tex]:
[tex]\[
x \approx 2.71828^6 \approx 403.4287934927351
\][/tex]
Thus, the solution to the equation [tex]\(\ln(x) = 6\)[/tex] is approximately [tex]\(x \approx 403.4287934927351\)[/tex].
