91Ó°ÊÓ

The natural logarithm, often denoted as \(\text{ln}\), is a logarithm to the base of the mathematical constant \(e\) (approximately equal to 2.71828).
In simpler terms, the natural logarithm answers the question: 'To what power must the base \(e\) be raised, to obtain a given number?' For instance, if you have \(e^x = 5\), the natural logarithm of 5 can be expressed as \(x = \text{ln}(5)\).
In the given function \(f(x, y) = \text{ln}(x) + y^3\), \( \text{ln} (x) \) specifically denotes the natural logarithm component. Handling logarithms becomes simpler when you understand key properties:

• \(\text{ln}(1) = 0\)
• \(\text{ln}(e) = 1 \)
• \(\text{ln}(e^k) = k\)

When evaluating the function for different values of \(x\), remembering these properties is useful to simplify calculations.

Function evaluation involves substituting specific values into the function and simplifying.
Here, we consider the function \(f(x, y) = \text{ln}(x) + y^3 \). Let's break down the steps to find \(f(e, 2)\), \(f(e^2, 4)\) and \(f(e^3, 5)\).

• **Finding \(f(e, 2)\):**
  Substitute \(x = e\) and \(y = 2\). Using \( \text{ln}(e) = 1 \), we get
  \( f(e, 2) = \text{ln}(e) + 2^3 = 1 + 8 = 9 \).
• **Finding \(f(e^2, 4)\):**
  Substitute \(x = e^2 \) and \(y = 4\). Using \( \text{ln}(e^2) = 2 \), we get
  \( f(e^2, 4) = \text{ln}(e^2) + 4^3 = 2 + 64 = 66 \).
• **Finding \(f(e^3, 5)\):**
  Substitute \(x = e^3\) and \(y = 5\). Using \( \text{ln}(e^3) = 3 \), we get
  \( f(e^3, 5) = \text{ln}(e^3) + 5^3 = 3 + 125 = 128 \).

By following these steps, you can handle most function evaluations with ease.

Exponents can be understood as a mathematical operation, written as \(x^n\), which means multiplying \(x\) by itself \(n\) times.
They are powerful in expressing large numbers in a compact form and arise frequently in algebra and calculus.
In the function \(f(x, y) = \text{ln}(x) + y^3\), the exponent part belongs to \(y^3\), which means \(y\) raised to the power of 3.
Key points to remember about exponents include:

• The standard form \( x^3 = x \times x \times x \).
• Any number raised to the power of 1 remains the number itself: \( x^1 = x \).
• Any number raised to the power of 0 is 1: \( x^0 = 1 \) (except \(0^0\)).

In this function, when calculating cubes:

• 2^3 = 2 \times 2 \times 2 = 8
• 4^3 = 4 \times 4 \times 4 = 64
• 5^3 = 5 \times 5 \times 5 = 125

Understanding these basics helps in simplifying and solving expressions involving exponents.