Skip to main content
header search
ahihi OpenLearn
OpenLearn
Menu
sticky search

About this free course

Become an OU student

Download this course

Share this free course

Course content Course content
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.

More free courses

4 Change of base

Suppose you have logax and you want to find logbx.

Let log sub b x equals n so that x equals b super n

Taking logarithms to base a gives:

multiline equation row 1 log sub a x equals log sub a left parenthesis b super n right parenthesis row 2 Blank equals n log sub a b

Rearranging this, gives:

n equals log sub a x divided by log sub a b

So,   multiline equation row 1 log sub b x equals one divided by log sub a b times prefix multiplication of log sub a x a times s postfix times n equals log sub b x

Thus, if you want to change between natural logarithms and logarithms to the base 10, you can use the following:

log x equals natural log x divided by natural log 10
natural log x equals log x divided by log e

Try putting this idea into practice now.

Activity 5 Finding logs

Timing: Allow about 4 minutes

Find log9(x) given that log3x = 12

Hint: Can you express any of the numbers in the form 3n.

Answer

Using   multiline equation row 1 log sub b x equals one divided by log sub a b times prefix multiplication of log sub a x

multiline equation row 1 cap c times h times a times n times g times i times n times g postfix times t times h times e postfix times b times a times s times e postfix times g times i times v times e times s postfix times log sub nine x equals log sub three x divided by log sub three nine row 2 Blank equals 12 divided by log sub three three squared row 3 Blank equals multiline equation line 1 12 divided by two row 4 Blank equals six

Remember that  logaa = 1, so log33 = 1

You may well have not come across logarithms in this way before, so don’t worry if you needed to take your time to work your way through this week. Practice is so often the key to success with maths, so the next section is your chance to check your understanding and application of these ideas.