Mathematics for science and technology
Mathematics for science and technology

Start this free course now. Just create an account and sign in. Enrol and complete the course for a free statement of participation or digital badge if available.

Free course

Mathematics for science and technology

3 Natural logarithms

The most frequently used bases for logarithms are 10 and the number ‘e’. Rather like pi, the irrational number ‘e’ occurs frequently in many branches of mathematics and its applications to science and engineering. Logarithms to base 10 are known as common logarithms and those to base ‘e’ are called natural or Napierian logarithms after the mathematician who discovered them.

Natural logarithms have the property that loge ex = x.

Natural logarithms are used to solve equations that contain the exponential function ex where e is the irrational number 2.718281828 correct to ten significant figures. A graph of this function is shown in Figure 2.

Described image
Figure 2 The exponential function ex

To work with natural logs on a calculator there is usually a button labelled loge or ln, with the inverse ex often accessed by the 2nd function key.

The value of the constant e can easily be found by calculating the value of e1.

An example of an exponential function in science is radioactive decay. The half-life of a radioactive element is a constant value. This means that no matter how long the decay process has been continuing for it always takes the same time for the radioactivity to fall by half. If you compare Figure 2 to Figure 3 you should be able to see that they have similar shapes – although Figure 2 shows growth and Figure 3 decay.

Radioactive decay curve shown in a graph.
Figure 3 Radioactive decay curve

When you work with logarithms to base 10 it is convention to drop the subscript and just write log x. Natural logarithms are written as In x. This means that the rules for logarithms, from section 2, can be written as follows for natural logarithms.

Rule one natural log equation left hand side x times y equals right hand side natural log x postfix plus natural log y
Rule two natural log equation left hand side x divided by y equals right hand side natural log x postfix minus natural log y
Rule three natural log x super r equals r natural log x

Activity 4 Solve the equations

Timing: Allow about 7 minutes

Solve the equations for x, showing your answers using natural logs. For example, 5 ln 3.

Remember that  ln ex = x.

  1. 15 = 3e2x
  2. 2ex/10 + 16 = 20

Discussion

  1. 15 = 3e2x

    Divide both sides by 3

    5 = e2x

    multiline equation row 1 natural log of five equals natural log of open e super two times x close row 2 equals two times x natural log of e postfix times

    So,  ln 5 = 2x  as  ln e = 1

    x equals natural log of five divided by two

Remember that logaa = 1, so  ln e = 1.

  1. two e super negative x divided by 10 plus 16 equals 20

    Subtract 16 from both sides

    two e super negative x divided by 10 equals four

    Divide both sides by 2

    e super negative x divided by 10 equals two

    natural log of prefix of of open e super negative x divided by 10 close equals natural log of two

    So, negative x divided by 10 equals natural log of two

    Multiply both sides by –10

    x equals negative 10 natural log of two

The final thing that you need to be able to do with logarithms is change the base.

MST_1

Take your learning further

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses.

If you are new to University-level study, we offer two introductory routes to our qualifications. You could either choose to start with an Access module, or a module which allows you to count your previous learning towards an Open University qualification. Read our guide on Where to take your learning next for more information.

Not ready for formal University study? Then browse over 1000 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus371