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Logarithmic Functions

The inverse of exponentials — from ln(x) to log rules and real-world scales

If exponential functions ask "what do I get when I raise a base to this power?", then logarithms ask the reverse question: "what power do I need?" The logarithm is the inverse of the exponential — and their graphs are perfect mirror images of each other.

The two most common logarithms are the natural log ln(x) (base e) and the common log log(x) (base 10). Both share the same characteristic shape: they pass through (1, 0), climb slowly to the right, and have a vertical asymptote at x = 0.

In this lesson, you'll see the mirror relationship between ln(x) and e^x, compare natural and common logs, explore the powerful log rules that turn multiplication into addition, and discover why logarithmic scales appear everywhere from earthquake measurement to sound levels — with an AI tutor guiding you step by step.

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FAQ

What is a logarithm?
A logarithm answers the question: "What exponent do I need?" If b^y = x, then \log_b(x) = y. For example, \log_2(8) = 3 because 2^3 = 8. The logarithm is the inverse operation of exponentiation, just like subtraction is the inverse of addition.
What is the difference between log and ln?
ln(x) is the natural logarithm — it uses base e ≈ 2.718. log(x) usually means the common logarithm with base 10. They have the same shape but different scales: \ln(x) = \log(x) \times \ln(10) \approx 2.303 \times \log(x). In calculus, ln is more natural; in everyday measurement (pH, decibels), log base 10 is more common.
What are the main logarithm rules?
The three key log rules are: Product rule: \log(ab) = \log(a) + \log(b) — multiplication becomes addition. Power rule: \log(a^n) = n \cdot \log(a) — exponents become multipliers. Quotient rule: \log(a/b) = \log(a) - \log(b) — division becomes subtraction. These rules are why logarithms were historically used for computation before calculators existed.
Why are logarithms useful in the real world?
Logarithmic scales compress huge ranges of values into manageable numbers. The Richter scale measures earthquake energy (each whole number is 10× more energy). Decibels measure sound intensity logarithmically. The pH scale measures acidity as −log[H⁺]. In finance, logarithms help calculate how long it takes an investment to double. In computer science, algorithms with O(\log n) complexity (like binary search) are extremely efficient.

Related Topics

Exponential Growth & DecayQuadratic EquationsPower Functions