Understanding Logarithms

Logarithms are another way of writing indices.
The operation of finding a logarithm is the inverse operation to finding a power or exponent.
The equation y = log_b x means the same as b^y = x.
For instance, log base 10 of 1000 is three, written as log10 1000 = 3.

Properties of Logarithms

The Product Rule: log_b(mn) = log_b m + log_b n, this rule signifies that the logarithm of the product of two numbers is the sum of the logarithm of those numbers.
The Quotient Rule: log_b(m/n) = log_b m - log_b n, this rule means the logarithm of the division of two numbers is the difference of the logarithm of those numbers.
The Power Rule: log_b(m^n) = n ⋅ log_b m, this rule means the logarithm of a number raised to a power is the product of the power and the logarithm of the number.

A common logarithm is a logarithm with base 10. It is typically written as, log10 x or simply, log x.
A natural logarithm has a base e (approximately equal to 2.718). It is denoted as ln i.e., base e logarithm of x is written as ln x.

The Change of Base formula allows you to compare logarithms with different bases. The formula is as follows: log_b a = log_c a / log_c b.
This formula can be particularly useful when dealing with calculators, as they often can only handle base 10 (common logarithm) or base e (natural logarithm).

To solve a logarithmic equation, aim to rewrite it in exponential form, or use the properties of logarithms to simplify and solve.
Remember, only logarithms with the same base can be combined.