Logarithmic Functions (Cambridge O Level Additional Maths)

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2 May 2024

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Logarithmic Functions

What are logarithmic functions?

A logarithm is the inverse of raising to a power
If $a = b^{x}$ then $\log_{b} a = x$
- $a > 0$
- $b$ is called the base of the logarithm
Try to get used to ‘reading’ logarithm statements to yourself
- $\log_{. . .} (. . .) =$ would be read as “the power that you raise ... to, to get ..., is ”
- So $\log_{5} (125) = 3$ would be read as “the power that you raise 5 to, to get 125, is 3”

Connection between exponentials and logarithms

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 2^x = 8 we know that x must be 3
- Logarithms allow use to solve more complicated problems
  - For example, the equation 2^x = 10 does not have a clear answer
  - Instead, we can use our calculator to find the value of $\log_{2} 10$

How do I use logarithms?

Finding the values of logarithms

Recognising the rules of logarithms allows expressions to be simplified

Working out a complicated logarithm

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

Calculators can have, possibly, three different logarithm buttons

logarithm button with a base

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

The button for ln

Natural logarithms (see “e”)

The button for log (base 10)

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?

Abbreviations for common logarithms

10 is a common base
- log₁₀x is abbreviated to log x or lg x
The value e is another common base
- log_e 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"
- $\ln x \equiv \log_{e} x$
- It is important to remember that ln is a function and not a number
The natural logarithm ( $\ln x$ ) and the exponential function ( $e^{x}$ ) are inverses of each other
It is defined for all positive numbers ( $x > 0$ )
- $\ln x$ cannot be defined for negative numbers or $x = 0$

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

How can I solve equations involving e & ln?

The functions $e^{x}$ and $\ln x$ are inverses of each other
- If $e^{x} = a$ then $x = \ln a$
- If $\ln x = a$ then $x = e^{a}$
- 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 $f (x) = e^{k}$

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 e^xwhich can then be left as an exact value

Worked example

Solve the equation $e^{2 x} = 5$ , leaving your answer as an exact value.

Take the natural logarithm of both sides.

$\ln (e^{2 x}) = \ln 5$

Use the property that $\ln (e^{a}) = a$ .

$2 x = \ln 5$

Divide both sides by 2.

$\begin{array}{rcl} x & = & \frac{\ln 5}{2} \end{array}$

Do not use your calculator to evaluate this as the question asks for the answer given as an exact value.

$x = \frac{\ln 5}{2}$

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