# 11.9: Essential Skills 6

In this section, we describe natural logarithms, their relationship to common logarithms, and how to do calculations with them using the same properties of exponents.

### Natural Logarithms

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:

10^{1} = 10 = e^{2.303} ln *e*^{x} = *x* ln 10 = ln(*e*^{2.303}) = 2.303 log 10 = ln *e* = 1

According 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 10

Substituting any value *y* for 10 gives

ln *y* = 2.303 log *y*

Other 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 for *x* = 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 [*e*x]. 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*.

- 0.523
- 1.63

**Solution:**

- ln(0.523) = −0.648;
*e*^{−0.648}= 0.523 - ln(1.63) = 0.489;
*e*^{0.489}= 1.63

### Skill Builder ES2

What number is each value the natural logarithm of?

- 2.87
- 0.030
- −1.39

**Solution:**

- ln
*x*= 2.87;*x*=*e*^{2.87}= 17.6 = 18 to two significant figures - ln
*x*= 0.030;*x*=*e*^{0.030}= 1.03 = 1.0 to two significant figures - 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.

Operation | Exponential Form | Logarithm |
---|---|---|

Multiplication | (10^{a})(10^{b}) = 10^{a} ^{+} ^{b} | log(ab) = log a + log b |

(e^{x})(e^{y}) = e^{x} ^{+} ^{y} | ln(e^{x}e^{y}) = ln(e^{x}) + ln(e^{y}) = x + y | |

Division | ||

Inverse |

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.

### Skill Builder ES3

Calculate the natural logarithm of each number.

- 22 × 18.6
0.512.67 - 0.079 × 1.485
20.50.026

**Solution:**

- ln(22 × 18.6) = ln(22) + ln(18.6) = 3.09 + 2.923 = 6.01. Alternatively, 22 × 18.6 = 410; ln(410) = 6.02.
ln(0.512.67)=ln(0.51)−ln(2.67)=−0.67−0.982=−1.65. Alternatively,0.512.67=0.19; ln(0.19) = −1.66.- ln(0.079 × 1.485) = ln(0.079) + ln(1.485) = −2.54 + 0.395 = −2.15. Alternatively, 0.079 × 1.485 = 0.12; ln(0.12) = −2.12.
ln(20.50.026)=ln(20.5)−ln(0.026)=3.0204−(−3.65)=6.67. Alternatively,20.50.026=790; ln(790) = 6.67.

The answers obtained using the two methods may differ slightly due to rounding errors.

### Skill Builder ES4

Calculate the natural logarithm of each number.

- 34 × 16.5
2.100.052 - 0.402 × 3.930
0.16410.7

**Solution:**

- 6.33
- 3.70
- 0.457
- −4.178

### Contributors

- Anonymous