LogarithmsFoundation

The inverse of exponential growth — interactive Mathematics simulation for IIT-JEE.

Concept

$\log_b x$ answers one question: to what power must b be raised to get x? It converts multiplication into addition — the trick that ran science for 300 years of slide rules. The log curve is the exponential's mirror across y = x: passes (1, 0), climbs forever but ever slower, and plunges to −∞ as x → 0⁺.

Key formula

\log_b(xy) = \log_b x + \log_b y, \quad \log_b x^n = n\log_b x, \quad \log_b x = \frac{\ln x}{\ln b}

Derivation

All rules are exponent rules read backwards: $b^m \cdot b^n = b^{m+n}$ becomes the product rule.

Change of base: if $y = \log_b x$ then $b^y = x$ ; take ln of both sides: $y\ln b = \ln x$ . So EVERY log is just ln scaled — one curve family.

Scenarios to explore

Logarithms — log laws, base changes and the inverse of exponential growth.

Real-world applications

pH, decibels, Richter scale, stellar magnitudes — log scales everywhere.
Complexity: binary search is O(log n).
Compound interest solve-for-time: t = ln(A/P)/ln(1+r).

JEE exam tips

log_b a · log_a b = 1 — reciprocal pair.
Inequalities: base > 1 keeps direction; 0 < base < 1 FLIPS it.
a^(log_a x) = x and log_a(aˣ) = x — inverse cancellation both ways.

Common mistakes

log(x + y) ≠ log x + log y — the rule is for PRODUCTS.
Forgetting the domain: log of 0 or negatives is undefined (in ℝ).
(log x)² vs log x² — parentheses catastrophe.

Exam traps to avoid

log_b x with 0 < b < 1 is DECREASING — check the base before comparing.
Equations: always verify candidate solutions keep every argument positive.