Logaritmos

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Definition of Logarithm

log_b(x) = y means that b^y = x, where b > 0, b != 1, and x > 0.

Product Rule

log_b(M * N) = log_b(M) + log_b(N)

Quotient Rule

log_b(M / N) = log_b(M) - log_b(N)

Power Rule

log_b(M^k) = k * log_b(M)

Logarithm of 1

log_b(1) = 0 for any valid base b.

Logarithm of the Base

log_b(b) = 1 for any valid base b.

Identity Rule 1 (Exponentiation)

b^(log_b(x)) = x for x > 0.

Identity Rule 2 (Inverse)

log_b(b^x) = x for all real numbers x.

Change of Base Formula

log_b(x) = log_c(x) / log_c(b) for any valid base c.

Common Logarithm

A logarithm with base 10, written as log(x) without an explicit base.

Natural Logarithm

A logarithm with base e (Euler's number), written as ln(x).

Reciprocal Base Rule

log_(1/b)(x) = -log_b(x)

Reciprocal Argument Rule

log_b(1/x) = -log_b(x)

Base Power Rule

log_(b^k)(x) = (1/k) * log_b(x)

Power of Base and Argument

log_(b^k)(x^k) = log_b(x)

Logarithm Swap Rule

log_a(b) = 1 / log_b(a)

Exponent Base Swap Rule

a^(log_b(c)) = c^(log_b(a))

Logarithm Domain

The domain of f(x) = log_b(x) is x > 0 (all positive real numbers).

Logarithm Range

The range of f(x) = log_b(x) is (-infinity, +infinity) (all real numbers).

Derivative of Natural Log

d/dx [ln(x)] = 1 / x for x > 0.

Derivative of General Log

d/dx [log_b(x)] = 1 / (x * ln(b))

Integral of Natural Log

Integral of ln(x) dx = x * ln(x) - x + C

Logarithmic Differentiation

A technique taking the natural log of both sides of an equation to simplify differentiating complex products, quotients, or powers.

Monotonicity of Logarithms (b > 1)

If b > 1, f(x) = log_b(x) is strictly increasing: x1 < x2 implies log_b(x1) < log_b(x2).

Monotonicity of Logarithms (0 < b < 1)

If 0 < b < 1, f(x) = log_b(x) is strictly decreasing: x1 < x2 implies log_b(x1) > log_b(x2).