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Logarithms (DP IB Maths: AI HL)
Revision Note
Introduction to Logarithms
What are logarithms?
- A logarithm is the inverse of an exponent
- If then where a > 0, b > 0, a ≠ 1
- This is in the formula booklet
- The number a is called the base of the logarithm
- Your GDC will be able to use this function to solve equations involving exponents
- If then where a > 0, b > 0, a ≠ 1
- 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”
- Two important cases are:
-
- Where e is the mathematical constant 2.718…
- This is called the natural logarithm and will have its own button on your GDC
-
- Logarithms of base 10 are used often and so abbreviated to log x
-
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 GDCs to find the value of
- We can solve some of these by inspection
Examiner Tip
- Before going into the exam, make sure you are completely familiar with your GDC and know how to use its logarithm functions
Worked example
Solve the following equations:
i)
,
ii)
, giving your answer to 3 s.f.
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Laws of Logarithms
What are the laws of logarithms?
- Laws of logarithms allow you to simplify and manipulate expressions involving logarithms
- The laws of logarithms are equivalent to the laws of indices
- The laws you need to know are, given :
-
- This relates to
-
- This relates to
-
- This relates to
-
- These laws are in the formula booklet so you do not need to remember them
- You must make sure you know how to use them
Useful results from the laws of logarithms
- Given
-
- This is equivalent to
-
- If we substitute b for a into the given identity in the formula booklet
- where
- gives
- This is an important and useful result
- Substituting this into the third law gives the result
- Taking the inverse of its operation gives the result
- From the third law we can also conclude that
- These useful results are not in the formula booklet but can be deduced from the laws that are
- Beware…
- …
- These results apply to too
- Two particularly useful results are
- Two particularly useful results are
- Laws of logarithms can be used to …
- simplify expressions
- solve logarithmic equations
- solve exponential equations
Examiner Tip
- Remember to check whether your solutions are valid
- log (x+k) is only defined if x > -k
- You will lose marks if you forget to reject invalid solutions
Worked example
a)
Write the expression in the form , where .
b) Hence, or otherwise, solve .
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