1. The Fundamentals
Before learning the laws, remember that a logarithm is simply the inverse of an index (power). If you can remember this relationship, the laws will make much more sense.
If \( a^x = n \), then \( \log_a(n) = x \)
The Three Core Laws
These three laws allow us to manipulate and solve complex equations involving powers.
Multiplication Law
\( \log_a(xy) \)
=
\( \log_a(x) + \log_a(y) \)
Division Law
\( \log_a\left(\frac{x}{y}\right) \)
=
\( \log_a(x) - \log_a(y) \)
Power Law
\( \log_a(x^k) \)
=
\( k\log_a(x) \)
Special Identities
- Log of 1: \( \log_a(1) = 0 \) (because \( a^0 = 1 \))
- Log of Base: \( \log_a(a) = 1 \) (because \( a^1 = a \))
Test Your Knowledge
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