Logarithms

Logarithm

The power to which a base number must be raised to obtain a given value.

Exponential Form

An expression in the format b^x = y, where b is the base, x is the exponent, and y is the result.

Logarithmic Form

An expression written as log_b(y) = x, indicating that b raised to the power of x equals y.

Base

The number that is multiplied by itself in an exponential expression.

Argument

The target number in a logarithmic expression.

Common Logarithm

Logarithm with base 10, denoted as log(x) or log_10(x).

Natural Logarithm

Logarithm with base e, denoted as ln(x).

Product Rule

The logarithm of a product is equal to the sum of the logarithms of the factors: logb(xy) = logb(x) + log_b(y).

Quotient Rule

The logarithm of a quotient is equal to the difference of the logarithms: logb(x/y) = logb(x) - log_b(y).

Power Rule

The logarithm of a number raised to a power is the power multiplied by the logarithm of the number: logb(x^p) = p * logb(x).

Change of Base Formula

A method to calculate logarithms of bases not available on calculators: logb(x) = logc(x)/log_c(b) for any base c.

Special Logarithm Value: log_b(1)

Always equals 0, because any base raised to the power of 0 gives 1.

Special Logarithm Value: log_b(b)

Always equals 1, as any base raised to the power of 1 equals itself.

Special Logarithm Value: b^(log_b(x))

Equals x, showing that exponential and logarithmic functions with the same base cancel each other out.

Special Logarithm Value: log_b(b^x)

Equals x, confirming the cancellation between logarithmic and exponential forms.