Worked example 1
Convert forms
Write 2⁵ = 32 in logarithmic form
- The base is 2.
- The result is 32.
- The exponent is 5.
Answer: log₂ 32 = 5
Higher Mathematics
Higher Maths Exponential and logarithmic functions
Practise exponential and logarithmic functions for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Connect exponential and logarithmic functions as inverses and interpret their graphs.
Before you start
- Know index laws.
- Understand inverse operations.
- Remember that log inputs must be positive.
Explanation
Exponential functions model repeated multiplication and often describe growth or decay. Logarithmic functions reverse exponentials: if aˣ = b, then logₐ b = x.
Their graphs are linked as inverses. Exponential graphs have positive outputs, while log graphs have positive inputs only.
Method and rules
- aˣ = b is equivalent to logₐ b = x.
- eˣ and ln x are inverse functions.
- For ln x, the domain is x > 0.
- Growth has multiplier greater than 1; decay has multiplier between 0 and 1.
Worked examples
Worked example 2
Use ln as an inverse
Solve eˣ = 7.
- Apply ln to both sides.
- Use ln(eˣ) = x.
So: x = ln 7
Watch out
- Treating log(a + b) as log a + log b.
- Taking logs of zero or negative expressions.
- Forgetting that eˣ is always positive.
- Mishandling powers when converting forms.
Exam reminder
When solving log equations, check every final solution in the original log arguments.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.