What is the relationship between exponential growth and logarithms?

Exponential growth and logarithms are inverse functions; logarithms undo the effect of exponential growth.

Exponential growth occurs when a quantity increases by a constant percentage over equal time intervals. For example, if you have £100 and it grows by 10% each year, after one year you would have £110, after two years £121, and so on. This type of growth can be described by the formula \( y = a \cdot b^x \), where \( a \) is the initial amount, \( b \) is the growth factor, and \( x \) is the number of time intervals.

Logarithms, on the other hand, are used to find the exponent that a base number must be raised to in order to get a certain value. For instance, if you want to know how many years it will take for your £100 to grow to £200 at a 10% annual growth rate, you would use a logarithm. The logarithmic function is written as \( \log_b(y) = x \), where \( b \) is the base, \( y \) is the result, and \( x \) is the exponent.

In simpler terms, if exponential growth tells you how much something grows over time, logarithms help you work backwards to find out how long it takes to reach a certain amount. For example, if \( y = 2^x \) describes exponential growth, then \( x = \log_2(y) \) would be the corresponding logarithmic function. This relationship is crucial in many areas of mathematics and science, as it allows us to solve problems involving growth and decay, such as population growth, radioactive decay, and interest calculations.