~3 min read • Updated Feb 11, 2026
1. What Is a Logarithm?
A logarithm tells us the exponent needed to obtain a number from a given base. The formal definition is:
log_a(b) = x means a^x = b
Here, a is the base, b is the number, and x is the logarithm.
Example:
log_2(8) = 3 because 2^3 = 8
2. What Is a Logarithmic Function?
A logarithmic function is the inverse of an exponential function and is defined as:
f(x) = log_a(x)
where a is a positive real number not equal to 1.
3. Properties of Logarithmic Functions
- Increasing when
a > 1 - Decreasing when
0 < a < 1 - Domain:
(0, ∞) - Range:
(-∞, ∞) - Always passes through
(1, 0) - Inverse of the exponential function
4. Graph of a Logarithmic Function
The graph of f(x) = log_a(x) has the following characteristics:
- Approaches the y-axis but never touches it
- Positive for
x > 1 - Negative for
0 < x < 1 - Crosses the x-axis at
(1, 0)
5. Fundamental Logarithmic Identities
1. Product Rule
log_a(xy) = log_a(x) + log_a(y)
2. Quotient Rule
log_a(x / y) = log_a(x) - log_a(y)
3. Power Rule
log_a(x^k) = k · log_a(x)
4. Logarithm of 1
log_a(1) = 0
5. Logarithm of the Base
log_a(a) = 1
6. Change of Base Formula
log_a(x) = log_b(x) / log_b(a)
6. Natural Logarithm and Common Logarithm
Natural Logarithm (ln)
Logarithm with base e:
ln(x) = log_e(x)
Common Logarithm (log)
Logarithm with base 10:
log(x) = log_10(x)
7. Derivatives of Logarithmic Functions
Derivative of the Natural Logarithm:
d/dx [ln(x)] = 1/x
Derivative of Logarithm with Any Base:
d/dx [log_a(x)] = 1 / (x ln(a))
8. Applications of Logarithms
- Solving exponential equations
- Modeling growth and decay
- Logarithmic scales (Richter, decibel, pH)
- Data compression and analysis
- Probability and statistics
- Computer science (time complexity: log n)
9. Important Examples
Example 1:
log_2(32) = 5
because 2^5 = 32
Example 2:
ln(e^4) = 4
Example 3:
log(0.01) = -2
because 10^-2 = 0.01
10. Conclusion
Logarithms and logarithmic functions are essential tools in mathematics and science. They are the inverse of exponential functions and follow powerful identities such as product, quotient, power, and change of base rules. Logarithms are widely used in modeling, data analysis, physics, economics, and computer science.
Written & researched by Dr. Shahin Siami