How the Expanding Logarithms Calculator Works
Expanding a logarithm means rewriting a single log of a product, quotient, or power into a sum, difference, or scaled log of its simpler pieces. For an expression of the form log_b(A·B/C^n), the expanded form is log_b(A) + log_b(B) − n·log_b(C). These three log rules follow directly from the corresponding rules for exponents, so the expanded sum is not an approximation of the original expression — it is algebraically identical to it.
Formula and method
Three rules do all the work. The product rule, log_b(MN) = log_b(M) + log_b(N), comes from b^x·b^y = b^(x+y). The quotient rule, log_b(M/N) = log_b(M) − log_b(N), comes from b^x/b^y = b^(x−y). The power rule, log_b(M^p) = p·log_b(M), comes from (b^x)^p = b^(xp). Applying all three to log_b(A·B/C^n) — a product of A and B, divided by C raised to the n-th power — gives log_b(A) + log_b(B) − n·log_b(C). This calculator evaluates the original single logarithm and the expanded sum independently (using log_b(x) = ln(x)/ln(b)) and compares them, so the "Verification" result confirms the algebra numerically rather than just symbolically.
Common mistakes when expanding logarithms
- Splitting a sum inside the log: log_b(A + B) is not log_b(A) + log_b(B). There is no rule for the logarithm of a sum — only products, quotients, and powers can be expanded.
- Misplacing the exponent: the power rule only applies to an exponent on the entire argument of a log, e.g. log_b(C^n) = n·log_b(C). It does not mean (log_b C)^n.
- Turning division into division of logs: log_b(A/C) becomes log_b(A) − log_b(C), a subtraction — not log_b(A) divided by log_b(C).
- Ignoring the domain: the base b must satisfy b > 0 and b ≠ 1, and every argument (A, B, and C here) must be strictly positive, or the original logarithm is undefined even though the expanded algebra "looks" fine on paper.
Verifying by condensing back
You can check any expansion by running the rules in reverse — this is called condensing. Combine a sum of logs into the log of a product, a difference into the log of a quotient, and a coefficient in front of a log into an exponent inside it: log_b(A) + log_b(B) − n·log_b(C) condenses back to log_b(A·B/C^n). If condensing your expanded answer returns the original expression, the expansion is correct.
Real-world applications
- Simplifying expressions in calculus before differentiating, especially with logarithmic differentiation of products and powers.
- Working with logarithmic scales such as pH, decibels, and the Richter scale, all of which are defined as logs of ratios.
- Solving exponential equations by taking a logarithm of both sides and expanding the result into linear terms.
- Analyzing algorithmic complexity in computer science, where expressions like log(n·m) or log(n^k) are routinely expanded to isolate variables.