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Logarithmic Functions (Cambridge O Level Additional Maths)
Revision Note
Logarithmic Functions
What are logarithmic functions?
- A logarithm is the inverse of raising to a power
- If then
- is called the base of the logarithm
- Try to get used to ‘reading’ logarithm statements to yourself
- would be read as “the power that you raise ... to, to get ..., is ”
- So would be read as “the power that you raise 5 to, to get 125, is 3”
- A logarithm is the inverse of raising to a power so we can use rules to simplify logarithmic functions
Why use logarithms?
- Logarithms allow us to solve equations where the exponent is the unknown value
- We can solve some of these by inspection
- For example, for the equation 2x = 8 we know that x must be 3
- Logarithms allow use to solve more complicated problems
- For example, the equation 2x = 10 does not have a clear answer
- Instead, we can use our calculator to find the value of
- We can solve some of these by inspection
How do I use logarithms?
- Recognising the rules of logarithms allows expressions to be simplified
- Recognition of common powers helps in simple cases
- Powers of 2: 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 =16, …
- Powers of 3: 30 = 1, 31 = 3, 32 = 9, 33 = 27, 34 = 81, …
- The first few powers of 4, 5 and 10 should also be familiar
- Calculators can have, possibly, three different logarithm buttons
- This button allows you to type in any number for the base
- Natural logarithms (see “e”)
- Shortcut for base 10 although SHIFT button needed
- Before calculators, logarithmic values had to be looked up in printed tables
What notation might I see with logarithms?
- 10 is a common base
- log10 x is abbreviated to log x or lg x
- The value e is another common base
- loge x is abbreviated to ln x
- (log x)2 ≠ log x2
Examiner Tip
-
Before going into the exam, make sure you are completely familiar with your calculator and know how to use its logarithm functions
ln x
What is ln?
- ln is a function that stands for natural logarithm
- It is a logarithm where the base is the constant "e"
- It is important to remember that ln is a function and not a number
- The natural logarithm () and the exponential function () are inverses of each other
- It is defined for all positive numbers ()
- cannot be defined for negative numbers or
What are the properties of ln?
- Using the definition of a logarithm you can see
- is only defined for positive x
How can I solve equations involving e & ln?
- The functions and are inverses of each other
- If then
- If then
- If then
- If then
- If your equation involves "e" then try to get all the "e" terms on one side
- If "e" terms are multiplied, you can add the powers
- You can then apply ln to both sides of the equation
- If "e" terms are added, try transforming the equation with a substitution
- For example: If then
- You can then solve the resulting equation (usually a quadratic)
- Once you solve for y then solve for x using the substitution formula
- If "e" terms are multiplied, you can add the powers
- If your equation involves "ln", try to combine all "ln" terms together
- Use the laws of logarithms to combine terms into a single term
- If you have then solve
- If you have then solve
Examiner Tip
- If you're working on the non-calculator exam paper you may need to leave answers as exact values so using lnx is a good idea
- This can be solved with ex which can then be left as an exact value
Worked example
Solve the equation , leaving your answer as an exact value.
Take the natural logarithm of both sides.
Use the property that .
Divide both sides by 2.
Do not use your calculator to evaluate this as the question asks for the answer given as an exact value.
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