- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Functions, graphs and inverses practice question 7
Find the exact value of the expression below, where is the natural logarithm.
Write and as powers of .
Then use and .
An exact fraction, or a decimal to three places.
Theory UBC Math 100 §4.6CLP-1 §0.6Key results
Worked solution
Strategy
A root is a fractional power, the log of a quotient is a difference of logs, and log eᵏ = k. With those three facts the expression is arithmetic.
Working
The cube root is the power 1/3, and the log of a quotient is a difference.
log eᵏ = k.
Add.
Answer
The answer is seventeen thirds.
Check
e⁵/∛e ≈ 148.413/1.3956 ≈ 106.34, whose log is about 4.667; √e ≈ 1.6487, and 2 log 1.6487 ≈ 1.000, so the sum is about 5.667 ≈ 17/3.
Where this goes wrong
Adding the log of the denominator, or reading ∛e as e³. The log of a quotient subtracts, and ∛e = e1/3.
Leans on
Laws of exponents, from Power functions and first sketches
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