91Ó°ÊÓ

The natural logarithm, often written as \(\text{ln} x\), is a logarithm to the base of Euler’s number \(e\). In other words, \(\text{ln} x\) answers the question: 'To what power must \(e\) be raised to obtain \(x\)?'. For example, if \(\text{ln} x = 2\), this means \(e^2 = x\). The natural logarithm is widely used in mathematics, especially in calculus and differential equations, because it has simple derivatives and integrals.
Here’s how the natural logarithm works step-by-step in our exercise:
- We start from the equation: \(5 \text{ln} x = -15\).
- By dividing both sides by 5, we isolate the natural logarithm term: \( \text{ln} x = -3\).
The beauty of natural logarithms lies in their relation to exponential functions, as we’ll see in the next section.

Exponentiation involves raising a number to the power of an exponent. In our exercise, we see exponentiation when we raise both sides of the equation to the power of \(e\). This step is crucial to solve the logarithmic equation. When we have \(\text{ln} x = -3\), we exponentiate both sides, giving us:
\(e^{\text{ln} x} = e^{-3}\).
Because \(e\) and \(\text{ln}\) are inverse functions, the left side simplifies to \(x\), resulting in:
\(x = e^{-3}\).
This transformation from a logarithmic form to an exponential form solves the equation and allows us to find the approximate value of \(x\). Exponentiation is a powerful tool used to manipulate and solve logarithmic equations by reversing the process of taking a logarithm.

Euler's number, denoted as \(\text{e}\), is an irrational number approximately equal to 2.718. It is a fundamental constant in mathematics, much like \(\text{π}\). The importance of \(\text{e}\) comes from its unique properties, especially in calculus and complex analysis. Here’s why it’s pivotal often:
- Euler's number appears naturally in many growth processes, such as population growth, radioactive decay, and interest calculations in finance.
- It is the base of natural logarithms, meaning any natural logarithmic function can be expressed in terms of \(\text{e}\).
In our exercise, we use \(\text{e}\) to transform the logarithmic equation \(\text{ln} x = -3\) to an exponential form, resulting in:
\(x = e^{-3}\).
Using a calculator, we approximate this value to be about 0.050 when rounded to three decimal places.
Understanding \(\text{e}\) and its properties provides a strong foundation in many fields of science and engineering, making it an essential concept to grasp.