Many natural phenomena exhibit an exponential rate of increase or decrease. Population growth is an example of an exponential rate of increase, whereas a runner’s performance may show an exponential decline if initial improvements are substantially greater than those that occur at later stages of training. Exponential changes are represented logarithmically by e^x, where e is an irrational number whose value is approximately 2.7183. The natural logarithm, abbreviated as ln, is the power x to which e must be raised to obtain a particular number. The natural logarithm of e is 1 (ln e = 1).

Some important relationships between base-10 logarithms and natural logarithms are as follows:

10¹ = 10 = e^2.303 ln e^x = x ln 10 = ln(e^2.303) = 2.303 log 10 = ln e = 1According to these relationships, ln 10 = 2.303 and log 10 = 1. Because multiplying by 1 does not change an equality,

ln 10 = 2.303 log 10Substituting any value y for 10 gives

ln y = 2.303 log yOther important relationships are as follows:

log A^x = x log A ln e^x = x ln e = x = e^{ln x}Entering a value x, such as 3.86, into your calculator and pressing the “ln” key gives the value of ln x, which is 1.35 forx = 3.86. Conversely, entering the value 1.35 and pressing “e^x” key gives an answer of 3.86.On some calculators, pressing [INV] and then [ln x] is equivalent to pressing [ex]. Hence

e^ln3.86 = e^1.35 = 3.86 ln(e^3.86) = 3.86

Skill Builder ES1

Calculate the natural logarithm of each number and express each as a power of the base e.

1. 0.523
2. 1.63

Solution:

1. ln(0.523) = ?0.648; e^?0.648 = 0.523
2. ln(1.63) = 0.489; e^0.489 = 1.63

Skill Builder ES2

What number is each value the natural logarithm of?

1. 2.87
2. 0.030
3. ?1.39

Solution:

1. ln x = 2.87; x = e^2.87 = 17.6 = 18 to two significant figures
2. ln x = 0.030; x = e^0.030 = 1.03 = 1.0 to two significant figures
3. ln x = ?1.39; x = e^?1.39 = 0.249 = 0.25

Calculations with Natural Logarithms

Like common logarithms, natural logarithms use the properties of exponents. 

The number of significant figures in a number is the same as the number of digits after the decimal point in its logarithm. For example, the natural logarithm of 18.45 is 2.9151, which means that e^2.9151 is equal to 18.45.