Logarithmic Functions

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Logarithmic functions

Logarithmic Functions Notes fig1, A Level & AS Maths: Pure revision notes

a = b^x and log _b a = x are equivalent statements
a > 0
b is called the base
Every time you write a logarithm statement say to yourself what it means
- log₃ 81 = 4
  “the power you raise 3 to, to get 81, is 4”
- log_p q = r
  “the power you raise p to, to get q, is r”

Logarithm rules

As a logarithm is the inverse of raising to a power

How do I use logarithms?

Logarithmic Functions Notes fig3, A Level & AS Level Pure Maths Revision Notes

Recognising the rules of logarithms allows expressions to be simplified

Logarithmic Functions Notes fig4, A Level & AS Maths: Pure revision notes

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
For more awkward cases a calculator is needed

Logarithmic Functions Notes fig5, A Level & AS Level Pure Maths Revision Notes

Calculators can have, possibly, different logarithm buttons

Logarithmic Functions Notes fig6, A Level & AS Maths: Pure revision notes

This button allows you to type in any number for the base

Logarithmic Functions Notes fig8, A Level & AS Maths: Pure revision notes

Shortcut for base 10 although SHIFT button needed
Before calculators, logarithmic values had to be looked up in printed tables

Notation

10 is a common base
- log₁₀x is abbreviated to log x or lg x
(log x)² ≠ log x²

Worked example

Logarithmic Functions Example fig1, A Level & AS Maths: Pure revision notes

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"ln"

What is ln?

ln is a function that stands for natural logarithm
It is a logarithm where the base is the constant "e"
- $\ln x \equiv \log_{e} x$
- It is important to remember that ln is a function and not a number

What are the properties of ln?

Using the definition of a logarithm you can see
- $\ln 1 = 0$
- $\ln e = 1$
- $\ln e^{x} = x$
- $\ln x$ is only defined for positive x
As ln is a logarithm you can use the laws of logarithms
- $\ln a + \ln b = \ln (a b)$
- $\ln a - \ln b = \ln (\frac{a}{b})$
- $n \ln a = \ln (a^{n})$

How can I solve equations involving e & ln?

The functions $e^{x}$ and $\ln x$ are inverses of each other
- If $e^{f (x)} = g (x)$ then $f (x) = \ln g (x)$
- If $\ln f (x) = g (x)$ then $f (x) = e^{g (x)}$
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
  - $e^{x} \times e^{y} = e^{x + y}$
  - 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 $y = e^{x}$ then $e^{4 x} = y^{4}$
  - You can then solve the resulting equation (usually a quadratic)
  - Once you solve for y then solve for x using the substitution formula
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 $\ln f (x) = \ln g (x)$ then solve $f (x) = g (x)$
- If you have $\ln f (x) = k$ then solve $f (x) = e^{k}$

Worked example

3-1-1-ln-we-solution

Examiner Tip

Always simplify your answer if you can
- for example, $\frac{1}{2} \ln 25 = \ln \sqrt{25} = \ln 5$
- you wouldn't leave your final answer as $\sqrt{25}$ so don't leave your final answer as $\frac{1}{2} \ln 25$

Logarithmic Functions (Edexcel International A Level Maths: Pure 2)

Revision Note