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
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Higher Mathematics
Exponential and logarithmic functions
Connect exponentials and logarithms as inverse functions 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.
Visual support
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.