- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
The chain rule and logarithms practice question 8
Logarithmic differentiation of the function below starts by writing as a sum. Which expression equals for ?
Which pieces of are multiplied, which are divided, and which are raised to a power? Check each choice against the log law for each kind of piece.
Here is the natural logarithm. Use , and . A square root is the power .
Theory UBC Math 100 §4.4CLP-1 §2.10Key results
Worked solution
Strategy
Use the log laws to turn the quotient into a difference, with both factors of the denominator subtracted, then bring every power down as a multiple. The logarithm of a sum, x² + 1, stays whole.
Working
A quotient becomes a difference, and a product in the denominator becomes two subtracted logarithms.
Powers come down as multiples. For x > 1/3 every factor is positive, so every logarithm is defined.
Answer
The answer is log f of x equals 4 log(x² + 1) minus one half log x minus 2 log(3x − 1).
Check
At x = 1, f(1) = 16/4 = 4, and this sum gives 4 log 2 − 2 log 2 = log 4, while adding the denominator’s logs would give log 64.
Where this goes wrong
Splitting log(x² + 1) into log(x²) + log 1, as if x² + 1 were a product. The log laws apply to products, quotients and powers; the logarithm of a sum has no rule and stays whole.
Leans on
Laws of logarithms, from Functions, graphs and inverses
Try a similar question
Use logarithmic differentiation to find for the function below.Saved in this browser only.