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Transforming Relationships to Linear Form (Cambridge O Level Additional Maths)
Revision Note
Transforming Relationships in the Form y=ax^n
How do I use logarithms to linearise a graph in the form y = axn?
- Logarithms can be used to linearise graphs of power functions
- Suppose
- You can take logarithms of both sides
- You can split the right hand side into the sum of two logarithms
- You can bring down the power in the final term
- You can take logarithms of both sides
- is in linear form
How can I use linearised form to find the unknown constants?
- After linearising the function it will be in the form
- is the gradient of the straight line graph
- is the -intercept of the straight line graph
- Once you know the value of the gradient of the straight line graph this is the value of
- You will need to find the value of by solving the equation
Examiner Tip
- You may need to leave your answer in exact form, especially in the non-calculator paper
- e.g. to solve ln a = 2 the answer will be a = e2
Worked example
The heights, metres, and the amount of time spent sleeping, hours, of a group of young giraffes can be modelled using , where and are constants.
The graph of against is a straight line passing through the points (1, -0.9) and (4, -4.5).
Find the values of and , giving your answers in exact form.
Find the gradient of the straight line between the two coordinates.
Substitute , and into the equation of a straight line ().
Substitute either coordinate in and rearrange to find .
Compare this to the linearised form of .
Take logarithms of both sides first and rearrange if you can't remember the correct form.
So is the value in front of and .
Solve by taking of both sides.
Transforming Relationships in the Form y=Ab^x
How do I use logarithms to linearise a graph in the form y = A(bx)?
- Logarithms can be used to linearise graphs of exponential functions
- Suppose
- You can take logarithms of both sides
- You can split the right hand side into the sum of two logarithms
- You can bring down the power in the final term
- You can take logarithms of both sides
- is in linear form
How can I use linearised form to find the unknown constants?
- After linearising the function it will be in the form
- is the gradient of the straight line graph
- is the -intercept of the straight line graph
- You will need to find the value of by solving the equation
- The value of will either be given or will need to be found
- You will need to find the value of by solving the equation
- The value of will either be given or will need to be found
Examiner Tip
- Unless the question specifies, you can choose whether to use ln, lg or log
- Remember you will need to solve the equation at the end
- If using lg, solve by taking 10 to the power of each side
- If using ln, solve by taking e to the power of each side
Worked example
Variables and are such that when is plotted against , a straight line passing through the points and is obtained.
Show that where and are constants to be found.
Find the gradient of the straight line between the two coordinates.
Substitute , and into the equation of a straight line ().
Substitute either coordinate in and rearrange to find .
Linearise .
Take logarithms of both sides first and rearrange if you can't remember the correct form.
Take logarithms of both sides.
Split the right-hand side into the sum of two logarithms.
Bring down the power in the final term.
Compare this to the equation of the line.
and
Solve by raising 10 to the power of both sides.
Solve by raising 10 to the power of both sides.
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