// Numbas version: exam_results_page_options {"name": "No feedback theme", "metadata": {"description": "

This exam uses a custom theme to provide no feedback about scores to the student.

The idea is to provide a version of the test compiled with this theme to the students as they attempt it. Once the test has closed, update with a version of the same test compiled with the default theme, so students can go back in and get feedback.

", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "questions": [{"name": "SFY0004 Logarithms1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": [], "advice": "\n

The rules for combining logs are

\\[\\begin{eqnarray*}&1.& \\log_b(ac)&=&\\log_b(a)+\\log_b(c)\\\\ \\\\ &2.& \\log_b\\left(\\frac{a}{c}\\right)&=&\\log_b(a)-\\log_b(c)\\\\ \\\\ &3.& \\log_b(a^r)&=&r\\log_b(a) \\end{eqnarray*} \\]

We see that:

\\[\\begin{eqnarray*}\\log_{\\var{b}}(\\var{a_1})-\\var{r_1}\\log_{\\var{b}}(\\var{a_2})+\\var{r_2}\\log_{\\var{b}}(\\var{a_3})&=&\\log_{\\var{b}}(\\var{a_1})-\\log_{\\var{b}}(\\var{a_2}^{\\var{r_1}})+\\log_{\\var{b}}(\\var{a_3}^{\\var{r_2}})\\mbox{ using 3.}\\\\&=&\\log_{\\var{b}}(\\var{a_1})-\\log_{\\var{b}}(\\var{a_2^r_1})+\\log_{\\var{b}}(\\var{a_3^r_2})\\\\&=&\\log_{\\var{b}}(\\var{a_1}\\times \\var{a_3^r_2})-\\log_{\\var{b}}(\\var{a_2^r_1}) \\mbox{ using 1.}\\\\&=&\\log_{\\var{b}}\\left(\\frac{\\var{a_1}\\times \\var{a_3^r_2}}{\\var{a_2^r_1}}\\right) \\mbox{ using 2.}\\\\&=&\\log_{\\var{b}}\\left(\\frac{\\var{a_1*a_3^r_2}}{\\var{a_2^r_1}}\\right)\\\\&=&\\log_{\\var{b}}\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)\\mbox{ on cancelling common factors}.\\end{eqnarray*}\\]

Hence $\\displaystyle c=\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}$.

To calculate $\\displaystyle \\log_{\\var{b}}\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)$ to 4 decimal places we use the fact that for any positive base $b$:

\\[\\log_b(c)=\\frac{\\ln(c)}{\\ln(b)}=\\frac{\\log_{10}(c)}{\\log_{10}(b)}\\]

and we can use either of the log functions, $\\ln$ or $\\log_{10}$ on our calculators to find the value.

Using $\\ln$ we find:

\\[ \\log_{\\var{b}}\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)=\\frac{\\ln\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)}{\\ln(\\var{b})}=\\var{ans}\\] to 4 decimal places.

\n ", "rulesets": {}, "parts": [{"prompt": "\n

Find a fraction or integer $c$ such that:

$\\log_{\\var{b}}(\\var{a_1})-\\var{r_1}\\log_{\\var{b}}(\\var{a_2})+\\var{r_2}\\log_{\\var{b}}(\\var{a_3})=\\log_{\\var{b}}(c)$

$c=\\;$[[0]].

Input $c$ as an integer or as a fraction and not as a decimal.

Now calculate $\\log_{\\var{b}}(c)$ to 4 decimal places:

$\\log_{\\var{b}}(c)=\\;$[[1]].

\n ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "all,fractionnumbers", "marks": 2.0, "answer": "{a_1*a_3^r_2}/{a_2^r_1}", "type": "jme"}, {"minvalue": "{ans-tol}", "type": "numberentry", "maxvalue": "{ans+tol}", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "

Answer the following question on logarithms.

", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"c": {"definition": "a_1*a_3^r_2/a_2^r_1", "name": "c"}, "b": {"definition": "random(2..9)", "name": "b"}, "a_3": {"definition": "random(2..8)", "name": "a_3"}, "a_2": {"definition": "random(2,4,8)*random(3,9)", "name": "a_2"}, "a_1": {"definition": "random(5..20)", "name": "a_1"}, "r_1": {"definition": "random(2..3)", "name": "r_1"}, "r_2": {"definition": "random(2..3)", "name": "r_2"}, "tol": {"definition": 0.0001, "name": "tol"}, "ans": {"definition": "precround(log(c)/log(b),4)", "name": "ans"}}, "metadata": {"notes": "\n \t\t

30/4/2013:

\n \t\t

Written for revision of logs in a foundation course

\n \t\t", "description": "\n \t\t

Given a sum of logs, all numbers are integers,

\n \t\t

$\\log_b(a_1)+\\alpha\\log_b(a_2)+\\beta\\log_b(a_3)$ write as $\\log_b(a)$ for some fraction $a$.

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Also calculate to 3 decimal places $\\log_b(a)$.

\n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Solving equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Steps", "algebra", "algebraic manipulation", "combining logarithms", "logarithm laws", "logarithms", "simplifying logarithms", "solving equations", "steps"], "advice": "\n

We use the following two rules for logs :

1. $\\log_a(b)-\\log_a(c)=\\log_a(b/c)$

2. $\\log_a(p)=r \\Rightarrow p=a^r$

Using rule 1 we get
\\[\\log_{\\var{a}}(x+\\var{b})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}\\left(\\simplify{(x+{b})/(x+{c})}\\right)\\]
So the equation to solve becomes:
\\[\\log_{\\var{a}}\\left(\\simplify{(x+{b})/(x+{c})}\\right)=\\var{d}\\]
and using rule 2 this gives:
\\[ \\begin{eqnarray} \\simplify{(x+{b})/(x+{c})}&=&{\\var{a}}^{\\var{d}}\\Rightarrow\\\\ x+\\var{b}&=&{\\var{a}}^{\\var{d}}(x+\\var{c})=\\simplify{{a^d}}(x+\\var{c})\\Rightarrow\\\\ \\simplify{{a^d-1}x}&=&\\simplify[std]{{b}-{c}*{a^d}={b-c*a^d}}\\Rightarrow\\\\ x&=&\\simplify{{b-c*a^d}/{a^d-1}} \\end{eqnarray} \\]
We should check that this solution gives positive values for $x+\\var{b}$ and $\\simplify{x+{c}}$ as otherwise the logs are not defined.

Substituting this value for $x$ into $\\log_{\\var{a}}(x+\\var{b})$ we get $\\log_{\\var{a}}(\\simplify{({b-c }{a^d})/{a^d-1}})$ so OK.

For $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get on substituting for $x$, $\\log_{\\var{a}}(\\simplify{({b-c })/{a^d-1}})$ so OK.

Hence the value we found for $x$ is a solution to the original equation.

\n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n

\\[\\log_{\\var{a}}(x+\\var{b})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\var{d}\\]

$x=\\;$ [[0]]

If you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.

Input all numbers as fractions or integers and not as decimals.

\n ", "gaps": [{"notallowed": {"message": "

Input as a fraction or an integer, not as a decimal.

", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "{b-c*a^d}/{a^d-1}", "type": "jme"}], "steps": [{"prompt": "\n

Two rules for logs should be used:

1. $\\log_a(b)-\\log_a(c)=\\log_a(b/c)$

2. $\\log_a(p)=r \\Rightarrow p=a^r$

Use rule 1 followed by rule 2 to get an equation for $x$.

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Solve the following equation for $x$.

Input your answer as a fraction or an integer as appropriate and not as a decimal.

\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..5)", "name": "a"}, "c": {"definition": "b-random(1..20)", "name": "c"}, "b": {"definition": "random(1..20)", "name": "b"}, "d": {"definition": "random(1,2)", "name": "d"}}, "metadata": {"notes": "\n \t\t

5/08/2012:

\n \t\t

Added tags.

\n \t\t

Added description.

\n \t\t

Checked calculation.OK.

\n \t\t

Improved display in content areas.

\n \t\t", "description": "

Solve for $x$: $\\log_{a}(x+b)- \\log_{a}(x+c)=d$

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Solving equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["Steps", "algebra", "algebraic manipulation", "combining logarithms", "logarithm laws", "logarithms", "simplifying logarithms", "solving", "solving equations", "steps"], "advice": "\n

We use the following three rules for logs :

1. $n\\log_a(m)=\\log_a(m^n)$

2. $\\log_a(b)-\\log_a(c)=\\log_a(b/c)$

3. $\\log_a(p)=r \\Rightarrow p=a^r$

Using rule 1 we get
\\[2\\log_{\\var{a}}(\\simplify{x+{b}})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}((\\simplify{x+{b}})^2)- \\log_{\\var{a}}(\\simplify{(x+{c})})\\]
Using rule 2 gives
\\[\\log_{\\var{a}}(\\simplify{(x+{b})^2})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\log_{\\var{a}}\\left(\\simplify{(x+{b})^2/(x+{c})}\\right)\\]
So the equation to solve becomes:
\\[\\log_{\\var{a}}\\left(\\simplify{(x+{b})^2/(x+{c})}\\right)=\\var{d}\\]
and using rule 3 this gives:
\\[ \\begin{eqnarray} \\simplify{(x+{b})^2/(x+{c})}&=&{\\var{a}}^{\\var{d}}\\Rightarrow\\\\ \\simplify{(x+{b})^2}&=&{\\var{a}}^{\\var{d}}(\\simplify{x+{c}})=\\simplify{{a^d}(x+{c})}\\Rightarrow\\\\ \\simplify{x^2+{2*b-a^(d)}x+{b^2-a^(d)*c}}&=&0 \\end{eqnarray} \\]
Solving this quadratic we get two solutions:

$x=\\var{sol1}$ and $x=\\var{sol2}$

We should check that these solutions gives positive values for $\\simplify{x+{b}}$ and $\\simplify{x+{c}}$ as otherwise the logs are not defined.

The value $x=\\var{sol1}$ gives:

Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{b}})$ we get $\\log_{\\var{a}}(\\simplify{{2*a^d}})$ so OK.

Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get $\\log_{\\var{a}}(\\simplify{{4*a^d}})$ so OK.

Hence $x=\\var{sol1}$ is a solution to our original equation.

The value $x=\\var{sol2}$ gives:

Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{b}})$ we get $\\log_{\\var{a}}(\\simplify{{-a^d}})$ so NOT OK.

Substituting this value for $x$ into $\\log_{\\var{a}}(\\simplify{x+{c}})$ we get $\\log_{\\var{a}}(\\simplify{{a^d}})$ so OK.

Hence $x=\\var{sol2}$ is NOT a solution to our original equation as $\\log_{\\var{a}}(\\simplify{x+{b}})$ is not defined for this value of $x$.

So there is only one solution $x=\\var{sol1}$.

\n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n

\\[2\\log_{\\var{a}}(\\simplify{x+{b}})- \\log_{\\var{a}}(\\simplify{(x+{c})})=\\var{d}\\]

$x=\\;$ [[0]].

If you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.

Input all numbers as fractions or integers and not as decimals.

\n ", "gaps": [{"notallowed": {"message": "

Input as an integer, not as a decimal.

Three rules for logs should be used:

1. $n\\log_a(m)=\\log_a(m^n)$

2. $\\log_a(b)-\\log_a(c)=\\log_a(b/c)$

3. $\\log_a(p)=r \\Rightarrow p=a^r$

So use rule 1 followed by rules 2 and 3 to get an equation for $x$.

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Solve the following equation for $x$.

Input your answer as a fraction or an integer as appropriate and not as a decimal.

\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2,3)", "name": "a"}, "c": {"definition": "b+2*a^(d)", "name": "c"}, "b": {"definition": "s*random(1..8)", "name": "b"}, "d": {"definition": "random(1,2)", "name": "d"}, "s": {"definition": "random(1,-1)", "name": "s"}, "sol2": {"definition": "-c+a^d", "name": "sol2"}, "sol1": {"definition": "c-2*b", "name": "sol1"}}, "metadata": {"notes": "\n \t\t

5/08/2012:

\n \t\t

Added tags.

\n \t\t

Added description.

\n \t\t

Checked calculation.OK.

\n \t\t

Improved display in content areas.

\n \t\t", "description": "\n \t\t

Solve for $x$: $\\displaystyle 2\\log_{a}(x+b)- \\log_{a}(x+c)=d$.

\n \t\t

Make sure that your choice is a solution by substituting back into the equation.

\n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "navigation": {"allowregen": false, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "never", "navigatemode": "sequence", "onleave": {"action": "warnifunattempted", "message": "You haven't submitted all your answers on this page."}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": false, "showtotalmark": false, "showanswerstate": false, "allowrevealanswer": false, "advicethreshold": 0, "intro": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}