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Mathematics for science and technology
Mathematics for science and technology

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4 Change of base

Suppose you have logax and you want to find logbx.

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

Taking logarithms to base a gives:

multiline equation row 1 log sub a of x equals log of sub a of open b super n close row 2 equals n log sub a of b of

Rearranging this, gives:

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

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

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

log of x equals natural log of x divided by natural log of 10
natural log of x equals log of x divided by log of 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 of x of equals one divided by log sub a of b times prefix multiplication of log of 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 of x of equals log sub three of x divided by log sub three of nine row 2 equals 12 divided by log sub three of three squared row 3 equals multiline equation line 1 12 divided by two row 4 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.