NATURAL LOGARITHMS

Definition Of Natural Logarithms

A logarithm to the base 'e' is called the Natural Logarithm (ln x), where 'e' is an irrational constant whose value is approximately equal to 2.71828.....

More About Natural Logarithms

The natural logarithm of 'e' itself is 1 because e¹ = e, and the natural logarithm of 1 is 0, since e⁰ = 1.
The natural logarithm function is the inverse function of the exponential function.

Video Examples: The Natural Logarithm

Examples of Natural Logarithms

The equation 24 = e^6x can be solved using natural logarithm.
Applying natural logarithm on both sides, ln 24 = ln e^6x
ln 24 = ln 6x, as ln e^ax = ax
x ∼ 0.53

Solved Example on Natural Logarithms

Ques: Solve the equation by using natural logarithms. 5^3x = 7^{(x + 1)}

Choices:

A. x ∼ 0.676
B. x ∼ 1.676
C. x ∼ 0.647
D. x ∼ 1.947
Correct Answer: A

Solution:

Step 1: Consider 5^3x = 7^{(x + 1)}
Step 2: Apply ln on both sides, ln 5^3x = ln 7^{(x + 1)}
Step 3: 3x ln 5 = (x + 1) ln 7
Step 4: 3x = (x + 1)
Step 5: 3x (0.827) = (x + 1)
Step 6: 2.48x = (x + 1)
Step 7: (2.48 - 1)x = 1
Step 8: 1.48 x = 1
Step 9: x ∼ 0.676

Quick Summary

ln(e) = 1
ln(1) = 0
Natural logarithm is the inverse function of the exponential function.

\[ ln(x) = y \Leftrightarrow e^y = x \]

🍎 Teacher Insights

Emphasize the relationship between exponential and logarithmic functions. Provide plenty of practice problems involving solving equations with natural logarithms.

🎓 Prerequisites

Exponents
Logarithms
Algebra

Check Your Knowledge

Q1: Solve for x: e^(2x) = 5

x ∼ 0.805 x ∼ 1.609 x ∼ 2.5 x ∼ 10

Q2: Solve for x: 5^(3x) = 7^(x+1)

x ∼ 0.676 x ∼ 1.676 x ∼ 0.647 x ∼ 1.947

Frequently Asked Questions

Q: What is the value of 'e'?
A: 'e' is an irrational constant approximately equal to 2.71828.

Q: How do I solve equations with natural logarithms?
A: Use the properties of logarithms to isolate the variable. Remember that ln(e^x) = x and e^(ln(x)) = x.