Before the midterm

Misapplying a logarithm rule

The habit: A multiple in the exponent becomes a power: e^{2 log 3} = (e^{log 3})² = 9.

9 practice questions are built to catch it, in 5 topics.

What it looks like

A question from the chain rule and logarithms, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

Logarithmic differentiation of the function below starts by writing $\log f(x)$ as a sum. Which expression equals $\log f(x)$ for $x > \tfrac{1}{3}$ ?

f(x) = \frac{\left(x^{2} + 1\right)^{4}}{\sqrt{x}\,(3x - 1)^{2}}

With the slip: $\displaystyle 4\log\left(x^{2}\right) + 4\log 1 - \tfrac{1}{2}\log x - 2\log(3x - 1)$
Worked through: $\displaystyle 4\log\left(x^{2} + 1\right) - \tfrac{1}{2}\log x - 2\log(3x - 1)$

x² + 1 is a sum, not a product, so log(x² + 1) does not split into log(x²) + log 1. Only its fourth power comes down.

See the worked solution

Where it comes up

Functions, graphs and inverses2 questions
Continuity1 question
The chain rule and logarithms4 questions
Related rates1 question
Exponential growth and decay1 question

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

01
Find the exact value of the expression below, where $\log$ is the natural logarithm. $\displaystyle \log\!\left(\frac{e^{5}}{\sqrt[3]{e}}\right) + 2\log\sqrt{e}$
Functions, graphs and inversesIntroductory
02
Logarithmic differentiation starts by differentiating $\log f(x)$ . Let $f$ be the function below, for $x > 0$ , where $\log$ is the natural logarithm. Find $\dfrac{d}{dx}\bigl(\log f(x)\bigr)$ . $\displaystyle f(x) = \frac{(x + 1)^{5}}{x^{2}\left(x^{2} + 3\right)^{3}}$
The chain rule and logarithmsIntroductory
03
A student solves the equation below as shown. Find the first line that is wrong. $\displaystyle \log(x + 2) + \log(x - 1) = \log 4$
Functions, graphs and inversesStandard
04
Use logarithmic differentiation to find $f'(2)$ for the function below. $\displaystyle f(x) = \frac{x^{3}(x - 1)^{2}}{x + 1}$
The chain rule and logarithmsStandard
05
Use logarithmic differentiation to find $f'(1)$ for the function below. $\displaystyle f(x) = \frac{x^{2}\sqrt{x + 3}}{(x + 2)^{3}}$
The chain rule and logarithmsStandard
06
Find the constants $a$ and $b$ that make $f$ continuous everywhere, and enter the product $ab$ . Here $\log$ is the natural logarithm.
ContinuityCapstone

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