Mathematics · 19 / 23
Exponential and logarithm: inverse operations
Final-year concepts · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
The exponential is positive and changes sums into products. The natural logarithm is defined for positive inputs and reverses exponentiation: ln(eˣ) = x and e^(ln a) = a for a > 0. Exponential change has a constant relative rate, not a constant absolute increase.
- 01eˣ is positive
- 02ln needs positivity
- 03Inverse operations
- 04Check the domain
WORKED EXAMPLE
From principle to reasoning
For eˣ = 5, apply ln: x = ln 5 ≈ 1.609. For ln(x) = 2, the positive input is e² ≈ 7.389.
Solve with domain conditions
ln(2x−3)=ln(x+1) needs x>1.5. Since ln is injective on positive inputs, 2x−3=x+1 gives x=4. Both arguments are 5. Domain conditions belong to the solution itself.
Solve a problem in this subject
- Define the objects
State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.
- Choose a transformation
Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.
- Show the reasoning
Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.
- Test and conclude
Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.
YOUR TURN
01 · Try before checking
Solve ln(x − 1) = 0.
One hint
Check the domain, then use e⁰.
Reveal and understand the solution
The domain requires x > 1. x − 1 = e⁰ = 1, so x = 2, which is in the domain.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
Solve eˣ = 5 over the reals.
A hint to get started
Natural logarithm is the inverse of the exponential.
Step-by-step solution
- Both sides are positive.
- Apply ln: x = ln(5).
- The increasing exponential gives exactly one solution.
Which step did you miss? Retry without help, then explain why the method works.
Solve ln(x − 2) = 0.
A hint to get started
Set the domain before exponentiating.
Step-by-step solution
- The domain requires x > 2.
- x − 2 = e⁰ = 1, so x = 3.
- 3 is permitted, and ln(1) = 0.
Which step did you miss? Retry without help, then explain why the method works.
A quantity follows Q(t) = 80e^(−0.2t). Find its half-life.
A hint to get started
Set Q(t) = 40 and isolate the exponential.
Step-by-step solution
- e^(−0.2t) = 1/2.
- −0.2t = ln(1/2) = −ln(2).
- t = ln(2)/0.2 ≈ 3.466 time units.
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
ln(a + b) is not ln a + ln b. Logs turn products into sums.
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
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