The following should be completed without the use of a calculator.
", "parts": [{"distractors": ["", "", "", "", "", ""], "choices": ["$b^c=a$
", "$b^a=c$
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"], "marks": 0, "stepsPenalty": "1", "prompt": "The definition of a logarithm says if $b$ and $a$ are positive and $b$ is not equal to 1, then $\\log_b(a)=c$ is equivalent to:
", "scripts": {}, "displayColumns": 0, "variableReplacements": [], "type": "1_n_2", "displayType": "radiogroup", "showFeedbackIcon": true, "showCorrectAnswer": true, "minMarks": 0, "steps": [{"type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "prompt": "The definition of a logarithm says if $b$ and $a$ are positive and $b$ is not equal to 1, then $\\log_b(a)=c$ is equivalent to $b^c=a$.
\n\nThis means to determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.
", "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": []}], "matrix": ["1", 0, 0, 0, 0, 0], "maxMarks": 0, "shuffleChoices": true, "variableReplacementStrategy": "originalfirst"}, {"distractors": ["", "", "", "", "", ""], "choices": ["$\\log_x (z)=y$
", "$\\log_x (y)=z$
", "$\\log_y (x)=z$
", "$\\log_y (z)=x$
", "$\\log_z (y)=x$
", "$\\log_z (x)=y$
"], "marks": 0, "stepsPenalty": "1", "prompt": "The definition of a logarithm says that if $x$ and $z$ are positive and $x$ is not equal to 1, then $x^y=z$ is equivalent to:
", "scripts": {}, "displayColumns": 0, "variableReplacements": [], "type": "1_n_2", "displayType": "radiogroup", "showFeedbackIcon": true, "showCorrectAnswer": true, "minMarks": 0, "steps": [{"type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "prompt": "The definition of a logarithm says if $b$ and $a$ are positive and $b$ is not equal to 1, then $\\log_b(a)=c$ is equivalent to $b^c=a$.
\n\nThis means to determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.
", "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": []}], "matrix": ["1", 0, 0, 0, 0, 0], "maxMarks": 0, "shuffleChoices": true, "variableReplacementStrategy": "originalfirst"}, {"gaps": [{"mustBeReduced": false, "marks": 1, "mustBeReducedPC": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{zero[1]}", "maxValue": "{zero[1]}", "scripts": {}, "variableReplacements": [], "type": "numberentry", "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "steps": [{"type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "prompt": "To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.
\n
To determine $\\log_{\\var{zero[0]}}(\\var{zero[0]^zero[1]})$, realise $\\var{zero[0]}^0=\\var{zero[0]^zero[1]}$ and so $\\log_{\\var{zero[0]}}(\\var{zero[0]^zero[1]})=0$.
Using the definition and your times tables determine the following:
\n$\\log_{\\var{zero[0]}}(\\var{zero[0]^zero[1]})$ = [[0]]
", "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": []}, {"gaps": [{"mustBeReduced": false, "marks": 1, "mustBeReducedPC": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{one[1]}", "maxValue": "{one[1]}", "scripts": {}, "variableReplacements": [], "type": "numberentry", "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "steps": [{"type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "prompt": "To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.
\n
To determine $\\log_{\\var{one[0]}}(\\var{one[0]^one[1]})$, realise $\\var{one[0]}^1=\\var{one[0]}$ and so $\\log_{\\var{one[0]}}(\\var{one[0]^one[1]})=1$.
Using the definition and your times tables determine the following:
\n$\\log_{\\var{one[0]}}(\\var{one[0]^one[1]})$ = [[0]]
", "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": []}, {"gaps": [{"mustBeReduced": false, "marks": 1, "mustBeReducedPC": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{two[1]}", "maxValue": "{two[1]}", "scripts": {}, "variableReplacements": [], "type": "numberentry", "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "steps": [{"type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "prompt": "To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.
\n
To determine $\\log_{\\var{two[0]}}(\\var{two[0]^two[1]})$, realise $\\var{two[0]}^2=\\var{two[0]^2}$ and so $\\log_{\\var{two[0]}}(\\var{two[0]^two[1]})=2$.
Using the definition and your times tables determine the following:
\n$\\log_{\\var{two[0]}}(\\var{two[0]^two[1]})$ = [[0]]
", "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": []}, {"gaps": [{"mustBeReduced": false, "marks": 1, "mustBeReducedPC": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{small[1]}", "maxValue": "{small[1]}", "scripts": {}, "variableReplacements": [], "type": "numberentry", "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "steps": [{"type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "prompt": "To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.
\n
To determine $\\log_{\\var{small[0]}}(\\var{small[0]^small[1]})$, realise $\\var{small[0]}^\\var{small[1]}=\\var{small[0]^small[1]}$ and so $\\log_{\\var{small[0]}}(\\var{small[0]^small[1]})=\\var{small[1]}$.
Using the definition and your times tables determine the following:
\n$\\log_{\\var{small[0]}}(\\var{small[0]^small[1]})$ = [[0]]
", "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": []}, {"gaps": [{"mustBeReduced": false, "marks": 1, "mustBeReducedPC": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{tens[1]}", "maxValue": "{tens[1]}", "scripts": {}, "variableReplacements": [], "type": "numberentry", "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "stepsPenalty": "1", "steps": [{"type": "information", "showFeedbackIcon": true, "showCorrectAnswer": true, "marks": 0, "prompt": "To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.
\n
To determine $\\log_{\\var{tens[0]}}(\\var{tens[0]^tens[1]})$, realise $\\var{tens[0]}^\\var{tens[1]}=\\var{tens[0]^tens[1]}$ and so $\\log_{\\var{tens[0]}}(\\var{tens[0]^tens[1]})=\\var{tens[1]}$.
Recall that $10^n$ is the same as a $1$ with $n$ zeros behind it.
", "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": []}], "prompt": "Using the definition and your times tables determine the following:
\n$\\log_{\\var{tens[0]}}(\\var{tens[0]^tens[1]})$ = [[0]]
", "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": []}], "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question"}, {"name": "Logs: resulting in negatives", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": ""}, "statement": "The following should be completed without the use of a calculator.
", "functions": {}, "tags": ["logarithms", "logs", "Logs", "negative indices", "negative powers"], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "", "variables": {"small": {"templateType": "anything", "definition": "random([2,3],[2,4],[3,3],[3,4],[4,3],[5,3])", "name": "small", "description": "", "group": "Ungrouped variables"}, "num1": {"templateType": "anything", "definition": "random(10..90)", "name": "num1", "description": "", "group": "Ungrouped variables"}, "num2": {"templateType": "anything", "definition": "random(2..12)", "name": "num2", "description": "", "group": "Ungrouped variables"}, "one": {"templateType": "anything", "definition": "random(map([b,1],b,list(2..12)))", "name": "one", "description": "", "group": "Ungrouped variables"}, "tens": {"templateType": "anything", "definition": "random([10,3],[10,4],[10,5],[10,6])", "name": "tens", "description": "", "group": "Ungrouped variables"}, "two": {"templateType": "anything", "definition": "random(map([b,2],b,list(2..12)))", "name": "two", "description": "", "group": "Ungrouped variables"}}, "ungrouped_variables": ["one", "two", "small", "tens", "num1", "num2"], "parts": [{"stepsPenalty": "2", "gaps": [{"correctAnswerFraction": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "maxValue": "1/{num1}", "mustBeReduced": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": true, "minValue": "1/{num1}", "correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "type": "numberentry"}, {"correctAnswerFraction": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "maxValue": "1/{num2}^2", "mustBeReduced": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": true, "minValue": "1/{num2}^2", "correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "type": "numberentry"}], "variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "type": "information", "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "When learning index laws you would have seen that
\\[x^{-n}=\\frac{1}{x^n}.\\]
Indices and logs are intimately related, ensure you revise your index laws.
Recall that we can convert negative indices to fractions
\n$\\var{num1}^{-1}$ = [[0]]
\n$\\var{num2}^{-2}$ = [[1]]
", "scripts": {}}, {"stepsPenalty": "1", "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "maxValue": "-1", "mustBeReduced": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "minValue": "-1", "correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "type": "numberentry"}], "variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "type": "information", "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.
\n
To determine $\\log_{\\var{one[0]}}\\left(\\frac{1}{\\var{one[0]}}\\right)$, realise $\\var{one[0]}^{-1}=\\frac{1}{\\var{one[0]}}$ and so $\\log_{\\var{one[0]}}\\left(\\frac{1}{\\var{one[0]}}\\right)=-1$.
Using the definition and your times tables determine the following:
\n$\\log_{\\var{one[0]}}\\left(\\frac{1}{\\var{one[0]}}\\right)$ = [[0]]
", "scripts": {}}, {"stepsPenalty": "1", "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "maxValue": "-2", "mustBeReduced": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "minValue": "-2", "correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "type": "numberentry"}], "variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "type": "information", "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.
\n
To determine $\\log_{\\var{two[0]}}\\left(\\frac{1}{\\var{two[0]^2}}\\right)$, realise $\\var{two[0]}^{-2}=\\frac{1}{\\var{two[0]^2}}$ and so $\\log_{\\var{two[0]}}\\left(\\frac{1}{\\var{two[0]^2}}\\right)=-2$.
Using the definition and your times tables determine the following:
\n$\\log_{\\var{two[0]}}\\left(\\frac{1}{\\var{two[0]^2}}\\right)$ = [[0]]
", "scripts": {}}, {"stepsPenalty": "1", "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "maxValue": "{-small[1]}", "mustBeReduced": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "minValue": "{-small[1]}", "correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "type": "numberentry"}], "variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "type": "information", "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.
\n
To determine $\\log_{\\var{small[0]}}\\left(\\frac{1}{\\var{small[0]^small[1]}}\\right)$, realise $\\var{small[0]}^{-\\var{small[1]}}=\\frac{1}{\\var{small[0]^small[1]}}$ and so $\\log_{\\var{small[0]}}\\left(\\frac{1}{\\var{small[0]^small[1]}}\\right)=-\\var{small[1]}$.
Using the definition and your times tables determine the following:
\n$\\log_{\\var{small[0]}}\\left(\\frac{1}{\\var{small[0]^small[1]}}\\right)$ = [[0]]
", "scripts": {}}, {"stepsPenalty": "1", "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "maxValue": "{-tens[1]}", "mustBeReduced": false, "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "allowFractions": false, "minValue": "{-tens[1]}", "correctAnswerStyle": "plain", "marks": 1, "scripts": {}, "type": "numberentry"}], "variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "type": "information", "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.
\n
To determine $\\log_{\\var{tens[0]}}(\\var{1/tens[0]^tens[1]})$, realise $\\var{tens[0]}^{-\\var{tens[1]}}=\\var{1/tens[0]^tens[1]}$ and so $\\log_{\\var{tens[0]}}(\\var{1/tens[0]^tens[1]})=-\\var{tens[1]}$.
Recall that $10^{-n}$ is the same as a decimal with zeros everywhere except a $1$ at the $n$th decimal place.
", "scripts": {}}], "type": "gapfill", "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "Using the definition and your times tables determine the following:
\n$\\log_{\\var{tens[0]}}(\\var{1/tens[0]^tens[1]})$ = [[0]]
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