Logarithms
The logarithm is the inverse of the exponential function. It shows how many times a number must be multiplied by itself to get another number.
Here, a is the base (positive number, cannot be 1), b is the number to be logged (positive), and c is the result, i.e., the exponent.
Important Identities
- For all a > 0, if a ≠ 1:
These identities help simplify logarithms and solve equations.
Special Logarithms
The base-10 logarithm is called the common logarithm, denoted by: log. The natural logarithm has base e (Euler's number ≈ 2.718), denoted by: ln.
Logarithmic Equations
Example of a simple logarithmic equation:
This means x is the number such that 2³ = x, so x = 8.
Relationship with Exponential Functions
The logarithm and exponential function are inverses of each other. This means the logarithm 'reverses' the exponentiation.
Practical Applications
- Finance: the relationship between exponential and logarithm appears in compound interest calculations.
- Sciences: the pH scale in chemistry is based on logarithms.
- Computer Science: it appears in measuring algorithm complexity (e.g., O(log n)).
Practice Exercise
We have reviewed and checked the materials, but errors may still occur. The content is provided for educational purposes only, so use it at your own responsibility and verify with other sources if needed.
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