## MATH1003 Quizzes

Quiz 5: Properties of Logs and Exponentials
Question 1 Questions
Which of the following are defined for all real values of $x$ ?
$\left(1\right)\phantom{\rule{1em}{0ex}}{log}_{10}{e}^{x}\phantom{\rule{2em}{0ex}}\left(2\right)\phantom{\rule{1em}{0ex}}{\left({x}^{2}+1\right)}^{x}\phantom{\rule{2em}{0ex}}\left(3\right)\phantom{\rule{1em}{0ex}}{\left({x}^{2}-1\right)}^{x}$
Exactly one option must be correct)
 a) (1) and (2) and (3) b) (2) and (3) only. c) (1) and (2) only. d) (1) and (3) only. e) None of them

Choice (a) is incorrect
Remember that ${a}^{b}={e}^{blna}$ and so ${a}^{b}$ makes sense only when $a>0$.
Choice (b) is incorrect
Remember that ${a}^{b}={e}^{blna}$ and so ${a}^{b}$ makes sense only when $a>0$.
Choice (c) is correct!
Choice (d) is incorrect
Remember that ${a}^{b}={e}^{blna}$ and so ${a}^{b}$ makes sense only when $a>0$.
Choice (e) is incorrect
Which option is a simplified version of the expression $ln\left(5{e}^{2x}\right)+{e}^{ln\left(5x\right)}$ ? Exactly one option must be correct)
 a) $ln10+lnx+5x$, for all $x>0$ b) $ln5+10x$, for all $x$ c) $ln2+lnx+ln5$, for all $x>0$ d) $7x+ln5$, for all $x$ e) None of the above.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is correct!
The expression only makes sense when $x>0$ and so the answer is $7x+ln5$, for all $x>0$
If $y=ln\left(\sqrt{{e}^{2x}+4}\right)$, what is $x$ in terms of $y$ ? Exactly one option must be correct)
 a) $x=\frac{1}{2}ln\left({e}^{2y}-4\right)$ b) $x={\left(ln\left({e}^{2y}-4\right)\right)}^{2}$ c) $x=\frac{1}{2}\left(2y-ln4\right)$ d) $x=\frac{1}{4}ln\left({e}^{y}-4\right)$ e) $x=\frac{1}{2}ln\left({e}^{2y}-2\right)$

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
If $y={3}^{x{e}^{x}}$, select the option which equals $\frac{dy}{dx}.$ Exactly one option must be correct)
 a) ${3}^{x{e}^{x}}\left(x+1\right){e}^{x}ln3$ b) $\left(x{e}^{x}+{e}^{x}\right)ln3$ c) $x{e}^{x}\phantom{\rule{0.3em}{0ex}}{3}^{x{e}^{x}-1}$ d) $x{e}^{x}{3}^{x+{e}^{x}-2}$ e) None of the above

Choice (a) is correct!
Choice (b) is incorrect
Try writing $y$ as an exponential, or alternatively, take logs of both sides and use logarithmic differentiation.
Choice (c) is incorrect
Try writing $y$ as an exponential, or alternatively, take logs of both sides and use logarithmic differentiation.
Choice (d) is incorrect
Try writing $y$ as an exponential, or alternatively, take logs of both sides and use logarithmic differentiation.
Choice (e) is incorrect
Use logarithmic differentiation to find $\frac{dy}{dx}$ when $y=\sqrt{{x}^{3}+{x}^{4}}\phantom{\rule{2.77695pt}{0ex}}\frac{{\left(ln\left(sinx\right)\right)}^{2}}{1+sinx}$ Exactly one option must be correct)
 a) $y\left(\frac{3{x}^{2}+4{x}^{3}}{2\left({x}^{3}+{x}^{4}\right)}+2cotx-\frac{cosx}{1+sinx}\right)$ b) $y\left(\frac{3{x}^{2}+4{x}^{3}}{\left({x}^{3}+{x}^{4}\right)}+\frac{cotx}{lnsinx}-\frac{cosx}{1+sinx}\right)$ c) $y\left(\frac{3{x}^{2}+4{x}^{3}}{2\left({x}^{3}+{x}^{4}\right)}+\frac{2cotx}{lnsinx}-\frac{1}{1+sinx}\right)$ d) $y\left(\frac{3{x}^{2}+4{x}^{3}}{2\left({x}^{3}+{x}^{4}\right)}+\frac{2cotx}{lnsinx}-\frac{cosx}{1+sinx}\right)$ e) None of the above

Choice (a) is incorrect
Taking natural logs of both sides of the equation for $y$ gives
$lny=\frac{1}{2}ln\left({x}^{3}+{x}^{4}\right)+2ln\left(lnsinx\right)\right)-ln\left(1+sinx\right).$
Now differentiate both sides with respect to $x$.
Choice (b) is incorrect
Taking natural logs of both sides of the equation for $y$ gives
$lny=\frac{1}{2}ln\left({x}^{3}+{x}^{4}\right)+2ln\left(lnsinx\right)\right)-ln\left(1+sinx\right).$
Now differentiate both sides with respect to $x$.
Choice (c) is incorrect
Taking natural logs of both sides of the equation for $y$ gives
$lny=\frac{1}{2}ln\left({x}^{3}+{x}^{4}\right)+2ln\left(lnsinx\right)\right)-ln\left(1+sinx\right).$
Now differentiate both sides with respect to $x$.
Choice (d) is correct!
Choice (e) is incorrect
Which values of $t$ satisfy simultaneously the pair of inequalities $\mid ln|t|\mid <1$ and $\sqrt{{t}^{2}}>1$ ? Exactly one option must be correct)
 a) All $t$ such that $1. b) All $t$ in the intervals $\left(-e,-1\right)$ or $\left(1,e\right)$. c) All $t$ such that $\frac{1}{e}. d) All $t\ge 1$. e) All $t$ such that $-e or $\frac{1}{e}

Choice (a) is incorrect
Remember that the solution set of $\sqrt{{t}^{2}}>1$ will contain negative as well as positive numbers!
Choice (b) is correct!
Choice (c) is incorrect
Remember that if $\mid ln|t|\mid <1$, then $-1
Choice (d) is incorrect
Remember that if $\mid ln|t|\mid <1$, then $-1 and that the solution set of $\sqrt{{t}^{2}}>1$ will contain negative as well as positive numbers!
Choice (e) is incorrect
Remember that if $\mid ln|t|\mid <1$, then $-1 and that the solution set of $\sqrt{{t}^{2}}>1$ will contain negative as well as positive numbers!
If $y={3}^{-x}-{2}^{2x}$ and $z={3}^{2x}+{2}^{-x}$, what is $yz$ ? Exactly one option must be correct)
 a) ${3}^{x}+{2}^{x}-{6}^{2x}+{6}^{-x}$ b) ${3}^{x}-{2}^{x}+{6}^{-x}$ c) ${6}^{-2x}-{6}^{4x}$ d) ${3}^{x}-{2}^{x}+{6}^{x}$ e) None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is correct!
In fact, $yz={3}^{x}-{2}^{x}+{6}^{-x}\left(1-{6}^{3x}\right)$.
Solve the equation $ln\left(\sqrt{{e}^{x}}-1\right)=0$ for $x$ and enter your answer correct to two decimal places.

Correct!
Taking exponentials gives $\sqrt{{e}^{x}}-1=1$, after which we obtain $\sqrt{{e}^{x}}=2,$ and then $x=ln4$.
Have you taken exponentials of each side of the equation? This gives $\sqrt{{e}^{x}}-1=1$, from which you can obtain the unique solution for $x$ after further manipulations.
Two of the following have identical derivatives. Tick the pair that do. (Zero or more options can be correct)
 a) $6x-{x}^{2}+cos2x+2{sec}^{2}x$ b) $-\frac{1}{2}sinxcosx-2\left(x-3\right)+12$ c) $\frac{1}{2}sin2x+2{tan}^{2}x-{\left(x-3\right)}^{2}$ d) $cosxsinx+{sec}^{2}x-{x}^{2}$ e) $2{sec}^{2}x+3+sinxcosx+6x-{x}^{2}$

There is at least one mistake.
For example, choice (a) should be False.
There is at least one mistake.
For example, choice (b) should be False.
There is at least one mistake.
For example, choice (c) should be True.
There is at least one mistake.
For example, choice (d) should be False.
There is at least one mistake.
For example, choice (e) should be True.
Correct!
1. False
2. False
3. True
4. False
5. True
The equation ${2}^{x}y+{y}^{3}x=1$ defines $y$ implicitly as a function of $x$ near the point with coordinates $\left(0,1\right)$. What is the value of $\frac{dy}{dx}$ at this point? Exactly one option must be correct)
 a) $2ln2$ b) $-1-ln2$ c) $\frac{1+2ln2}{2}$ d) $1-ln2$ e) $ln2-2$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect