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In 1614, Scottish mathematician John Napier published a method to turn exhausting multi-digit multiplication into simple addition, transforming navigation and astronomy forever: the logarithm.
Because a logarithm represents an exponent (the power a base bb must be raised to produce a given value), every exponent rule translates directly into a rule for logarithms.
The input value inside a logarithm is called the argument, and these algebraic rules dictate how arguments multiply, divide, and scale.
Why do these rules actually hold true every single time? Let's trace one back to basic algebra.

Why the Rules Work

To prove the product rule, let x=log⁡b(M)x = \log_b(M) and y=log⁡b(N)y = \log_b(N), which means in exponential form bx=Mb^x = M and by=Nb^y = N.
Multiplying both sides gives M⋅N=bx⋅by=bx+yM \cdot N = b^x \cdot b^y = b^{x+y}. Rewriting this back into logarithmic form gives log⁡b(M⋅N)=x+y=log⁡b(M)+log⁡b(N)\log_b(M \cdot N) = x + y = \log_b(M) + \log_b(N).
Now that we know why they work, how do we use these laws to pull complex expressions apart?

Expanding Logarithmic Expressions

Expanding means breaking a single complex logarithm into a sum, difference, or multiple of simpler logarithms where each argument is a single variable or number with no exponents.
Consider expanding log⁡3(x4yz5)\log_3\left(\frac{x^4 \sqrt{y}}{z^5}\right). First, convert all radical signs into fractional exponents so y=y1/2\sqrt{y} = y^{1/2}.