Properties of Logarithms

These flashcards cover fundamental definitions and properties related to logarithms, which will help in understanding the subject for the exam.

Logarithm

The exponent by which a base must be raised to produce a given number.

Natural Logarithm

Logarithm to the base e, where e is approximately equal to 2.718.

Common Logarithm

Logarithm to the base 10.

One-to-One Property

If logb(y) = logb(x), then x = y for b > 0 and b ≠ 1.

Property of One

Logarithm of 1 in any base equals 0.

Multiplication Property of Logarithms

logb(xy) = logb(x) + log_b(y).

Division Property of Logarithms

logb(x/y) = logb(x) - log_b(y).

Power Property of Logarithms

logb(x^r) = r * logb(x).

Inverse Property of Logarithms

b^(logb(x)) = x and logb(b^x) = x.

Change of Base Formula

logb(x) = loga(x) / log_a(b) for any base a.

Logarithmic Equation Check

Always verify solutions to ensure no logarithm is taken of a negative number or zero.

log4 64

The logarithm of 64 to the base 4 is 3.

ln(x + 2) - ln(4x + 3)

Using Division Property leads to relation between numerator and denominator.

ln(e^x)

The natural logarithm function returns x.

log_b(b)

Logarithm of base b to itself equals 1.