1. Which is ln(1)?
- 1
- e
- 0
Check the answer and reasoning
0
e⁰=1, so its natural logarithm is zero.
MATH / A2 level
Exponential and logarithmic functions
Logarithms reverse exponentiation and convert products into sums. Natural logarithms use base e and require positive real arguments.
Learning objectives
- Explain exponential and logarithmic functions using a concrete example.
- Apply the worked method and justify each step.
- Recognise the limitation described in the final check.
The big idea
Logarithms reverse exponentiation and convert products into sums. Natural logarithms use base e and require positive real arguments.
Exponential models describe constant proportional growth or decay. Linearising data with logarithms can estimate model constants, but the transformed error pattern also matters.
Equations and relationships
ln(ab)=ln a+ln b, a,b>0
Product law for natural logarithms.
Worked example
Solve e^(2x)=7.
Follow the solution
Take natural logs: 2x=ln 7, then divide by 2.
Investigate and explain
Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.
Use the prediction activity below; a local model appears only where it supports this topic.
Check your understanding
2. Solve e^(2x)=7.
- x = ln(7)/2
- x=7/2
- x=ln(7/2)
Check the answer and reasoning
x = ln(7)/2
Take natural logs: 2x=ln 7, then divide by 2.
3. Is ln(a+b)=ln a+ln b generally valid?
- Yes
- No
- Only if both are positive
Check the answer and reasoning
No
The logarithm law applies to products, not sums.