Let's learn about the natural logarithm with some examples in A-Level Maths!

**the Natural Log**

The natural logarithm is often written as ln which you may have noticed on your calculator.

**ln** **x** **= log**_{e} **x**

The symbol e symbolizes a special mathematical constant. It has importance in growth and decay problems. The logarithmic properties listed above hold for all bases of logs. If you see log x written (with no base), the natural log is implied. The number e can not be written exactly in decimal form, but it is approximately 2.718. Of course, all the properties of logs that we have written down also apply to the natural log. In particular,

**e**^{y} **=** **x**

**and**

**ln** **x** **=** **y**

are equivalent statements. We also have e^{0} = 1 and ln 1 = 0.

Example 1 :

e^{log}^{e} ^{a} = a

Example 2 :

e^{a} ^{loge} ^{x} = e ^(log_{e} x^{a} ) = x^{a}

Example 3 :

log_{e} e^{2y}

= 2y log_{e} e

= 2y

Example 4 :

log_{e} (x^{2}/5) = 2 log_{e} x −log_{e} 5

**Exercise**

1. Simplify the following

(a) log x^{2} −log xy + 4 log y

(b) ln(8x)^{1/2} + ln 4x^{2} −ln(16x)^{1/2}

(c) e^{6}e^{−6}

(d) 12e^{7} ÷6e^{2}

(e) ln e^{2}

(f) ln(e^{2} ln e^{3})

2. Find x in each of the following:

(a) ln x = 2.7

(b) ln(x + 1) = 1.86

(c) x = e^{9.8} ÷e^{7.6}

(d) 6.27 = e^{x}

(e) 4.12 = e^{−2x}

**Ans**

1. (a) ln xy^{3}

(b) 1/2 ln 8 + 2 ln x

(c) 1

(d) 2e^{5}

(e) 2

(f) 2 + ln 3

2. (a) 14.88

(b) 5.42

(c) 9.03

(d) 1.84

(e) −0.71

Drafted by Eunice (Maths)

References:

https://www.onlinemathlearning.com/natural-logarithm.html

https://www.google.com/url?sa=i&url=https%3A%2F%2Famsi.org.au%2FESA_Senior_Years%2FSeniorTopic3%2F3h%2F3h_2content_2.html&psig=AOvVaw0A2kTUZ62shQ4WIN81H0BZ&ust=1628628860817000&source=images&cd=vfe&ved=0CAsQjhxqFwoTCLj_xLvppPICFQAAAAAdAAAAABAs