Logarithms are mathematical functions that allow us to efficiently solve equations involving exponents and exponential growth. They are the inverse operation of exponentiation and can be used to convert exponential equations into more manageable form.

The logarithm of a number with a given base is the exponent to which the base must be raised to obtain that number. For example, in the logarithm with base 10, the logarithm of 100 is 2 because 10 raised to the power of 2 equals 100 (i.e., 10² = 100). By convention, the base 10 logarithm is known as the common logarithm and is often denoted as log₁₀.

Logarithms have numerous applications in various fields, including mathematics, science, engineering, and finance. They are particularly useful when dealing with large numbers, as logarithms allow for simplification and scaling down of calculations.

One essential property of logarithms is their ability to transform multiplication into addition. Specifically, the logarithm of a product is equal to the sum of the logarithms of the individual factors. This property, known as the product rule, can significantly simplify complex calculations.

Another important property is the power rule of logarithms, which states that the logarithm of a number raised to a power is equal to the product of that power and the logarithm of the number itself. This allows for the efficient computation of large exponents.

Overall, logarithms provide a powerful tool in mathematics and other disciplines for solving equations involving exponential growth and simplifying complex calculations involving large numbers.