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Before calculators existed, seventeenth-century astronomers spent weeks multiplying massive eight-digit planetary distances by hand. In 1614, Scottish mathematician John Napier published a breakthrough that changed everything: an algebraic trick that turned brutal multiplication problems into simple addition.
A logarithm is simply an exponent finder: the expression log⁑b(a)\log_b(a) asks the question, "bb raised to what power equals aa?" Because logs output exponents, they inherit the fundamental rules of exponents.
This relationship gives us the product property of logarithms: for any positive base bb where bβ‰ 1b \neq 1, and positive numbers MM and NN, log⁑b(MN)=log⁑b(M)+log⁑b(N)\log_b(MN) = \log_b(M) + \log_b(N).
How do we use this rule to break down complicated algebraic expressions into simpler, workable parts?

Expanding Logarithmic Expressions

To expand a logarithmic expression means to split a single log with multiplied factors inside into a sum of separate, simpler logarithms.
Let's expand log⁑3(9x)\log_3(9x). Because 99 and xx are multiplied inside the argument, the product property splits them apart: log⁑3(9)+log⁑3(x)\log_3(9) + \log_3(x).
Since 32=93^2 = 9, the term log⁑3(9)\log_3(9) simplifies cleanly to 22. The fully expanded and simplified expression is 2+log⁑3(x)2 + \log_3(x).
What if an expression already has multiple logs added together β€” can we run this exact machinery in reverse?

Condensing Logarithmic Expressions

To condense an expression means to combine multiple logarithms connected by addition into a single logarithm whose argument is the product of their inputs.