- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Functions, graphs and inverses practice question 4
Solve the equation below for .
The left side is a difference of two logarithms. Which law combines them into one, and what undoes a natural logarithm?
Here is the natural logarithm, base . Combine the left side into one logarithm with .
means . Solve for , and check that .
An exact expression, or a decimal to three places.
Theory UBC Math 100 §4.6CLP-1 §0.6Key results
Worked solution
Strategy
Combine the two logarithms into one, undo the logarithm with the exponential, and solve the equation that is left. Then check that the answer lies in the domain, x > 0.
Working
A difference of logarithms is the logarithm of a quotient.
log is the natural logarithm, so exponentiate with base e.
Clear the fraction and collect the x terms.
Both x and x + 1 are positive, so both logarithms are defined.
Answer
The answer is x equals 1 over (e minus 1), about 0.582.
Check
At x = 0.582, (x + 1)/x ≈ 2.718, which is e, so the difference of the logs is 1.
Where this goes wrong
Reading log as the base-10 logarithm and answering 1/9. In calculus log means the natural logarithm, so undoing it gives a power of e, not a power of 10.
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