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 §1.5Key resultsAssumed from precalculus
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.
The idea behind it
The functions, graphs and inverses lesson, a figure to move, in a few steps
Leans on
Laws of exponents, from Power functions and first sketches
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