AI Tutor
What if your calculator only had two working number keysβ€”22 and 33β€”yet you were tasked with calculating the exact logarithm of 7272, 0.750.75, or 108\sqrt{108}?
Because every composite number is built from prime factors, knowing just a few foundational values gives you the power to evaluate an infinite number of logarithms.

Deconstructing into Primes

Under the Prime Factorisation Strategy, we decompose any composite integer argument NN into its prime factors (N=2rβ‹…3s…N = 2^r \cdot 3^s \dots) before applying log laws, enabling substitution of base parameters like p=log⁑a(2)p = \log_a(2) and q=log⁑a(3)q = \log_a(3).
We expand these factors using the Product Rule, log⁑a(xy)=log⁑a(x)+log⁑a(y)\log_a(xy) = \log_a(x) + \log_a(y), which converts products of prime powers into sums of logarithmic terms.
Similarly, the Power Rule, log⁑a(xk)=klog⁑a(x)\log_a(x^k) = k\log_a(x), applies to integer, negative, and fractional indices (such as log⁑a(x)=12log⁑a(x)\log_a(\sqrt{x}) = \frac{1}{2}\log_a(x) and log⁑a(1x)=βˆ’log⁑a(x)\log_a\left(\frac{1}{x}\right) = -\log_a(x)).
Next, the Power Rule, log⁑a(xk)=klog⁑a(x)\log_a(x^k) = k\log_a(x), pulls exponents out front. This applies to integer, negative, and fractional indices (e.g., log⁑a(x)=12log⁑a(x)\log_a(\sqrt{x}) = \frac{1}{2}\log_a(x) and log⁑a(1x)=βˆ’log⁑a(x)\log_a\left(\frac{1}{x}\right) = -\log_a(x)).
Let's test this: if p=log⁑a(2)p = \log_a(2) and q=log⁑a(3)q = \log_a(3), we rewrite 7272 as 23Γ—322^3 \times 3^2, expanding log⁑a(23Γ—32)\log_a(2^3 \times 3^2) into 3p+2q3p + 2q.
What happens when your argument contains fractions or radicals instead of whole integers?

Roots, Quotients, and Fractions

When Handling Rational Arguments and radicals, rewrite expressions with roots as fractional exponents (for example, 183=(2β‹…32)1/3=21/3β‹…32/3\sqrt[3]{18} = (2 \cdot 3^2)^{1/3} = 2^{1/3} \cdot 3^{2/3}) before expanding.
The Quotient Rule, log⁑a(xy)=log⁑a(x)βˆ’log⁑a(y)\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y), enables expressing fractional arguments and rational numbers in terms of parameter differences.