// Numbas version: finer_feedback_settings
{"name": "Chemical Thermodynamics:  Logarithm Practice - General  ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Chemical Thermodynamics:  Logarithm Practice - General  ", "tags": [], "metadata": {"description": "<p>Public</p>", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "statement": "<p>Calculate $x$ in the following expressions (you should try to do this <strong>without</strong> using a calculator, if possible):</p>", "advice": "<p>You can think of a logarithm as a <strong>question</strong>, written symbolically - the expression:</p>\n<p>$\\mathrm{log}_x(y)$</p>\n<p>means \"$x$ raised to <strong>which power</strong> (or 'index') equals $y$ ?\".</p>\n<p></p>\n<p>The answer to this question is the value of our logarithm, so if $x^z=y$, then the answer to our question would be $z$, and so this would also be the value of our logarithm:</p>\n<p>$\\mathrm{log}_x(y)=z$ .</p>\n<p></p>\n<p></p>\n<p></p>\n<p>For the first problem, we must calculate the value of a logarithm:</p>\n<p>$x=\\mathrm{log_{_{\\space\\var{nat}}}(\\var{ans2})}$ .</p>\n<p></p>\n<p>To find this, we must simply answer the question \"$\\var{nat}$ raised to which power equals $\\var{ans2}$ ?\" .</p>\n<p>You will probably know that $\\var{nat}^{\\var{expo}}=\\var{ans2}$ , so the answer to our question is $\\var{expo}$ . Thus,</p>\n<p>$x=\\mathrm{log_{_{\\space\\var{nat}}}(\\var{ans2})}=\\var{expo}$ .</p>\n<p></p>\n<p></p>\n<p></p>\n<p>For the second problem, $x$ is not the value of a logarithm; it is the <strong>base</strong> of one:</p>\n<p>$\\mathrm{log}_{\\space x}\\mathrm{(\\var{bigint})=2}$ .</p>\n<p></p>\n<p>To find $x$, let's consider the question the logarithm represents: \"$x$ raised to which power equals $\\var{bigint}$ ?\" .</p>\n<p>As we are <strong>given the value</strong> of the logarithm, we <strong>know the answer</strong> to this question: $2$ .</p>\n<p>This is the \"power\" mentioned in the question, so we actually know that: \"$x$ raised to power of $2$ equals $\\var{bigint}$\", i.e. $x^2=\\var{bigint}$</p>\n<p>Thus, $x=\\sqrt{\\var{bigint}}=\\var{int}$ .</p>\n<p></p>\n<p></p>\n<p></p>\n<p>For the third problem, we must calculate the value of a logarithm:</p>\n<p>$x=\\mathrm{log}_{\\space y}\\mathrm{(1)}$ .</p>\n<p></p>\n<p>To find this, we must answer the question \"$y$ raised to which power equals $1$ ?\" .</p>\n<p>We don't know the value of $y$, but thankfully, the value of <strong>any number</strong> raised to the power of <strong>zero</strong> is always equal to $1$, meaning that $y^0=1$ ; so, the answer to our question is simply $0$ . Thus,</p>\n<p>$x=\\mathrm{log}_{\\space y}\\mathrm{(1)}=0$ .</p>\n<p></p>\n<p><strong>Note</strong>: This means that any logarithm with an <strong>argument of $1$</strong> will have a value of $0$ .</p>\n<p></p>\n<p>For the fourth problem, we must calculate the value of a logarithm:</p>\n<p>$x=\\mathrm{log}_{\\space y}(y)$ .</p>\n<p></p>\n<p>This problem relies on a similar trick to its predecessor.</p>\n<p>To find $x$, we must answer the question \"$y$ raised to which power equals $y$ ?\" .</p>\n<p>We don't know the value of $y$, but thankfully, the value of <strong>any number</strong> raised to the power of <strong>one</strong> is always equal to the original number, meaning that $y^1=y$ ; so, the answer to our question is simply $1$ . Thus,</p>\n<p>$x=\\mathrm{log}_{\\space y}(y)=1$ .</p>\n<p></p>\n<p><strong>Note</strong>: This means that any logarithm with an <strong>argument equal to the base</strong> will have a value of $1$ .</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"ans2": {"name": "ans2", "group": "Ungrouped variables", "definition": "{nat}^({expo})", "description": "", "templateType": "anything", "can_override": false}, "int": {"name": "int", "group": "Ungrouped variables", "definition": "random(1 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "nat": {"name": "nat", "group": "Ungrouped variables", "definition": "random(2 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}, "expo": {"name": "expo", "group": "Ungrouped variables", "definition": "random(2 .. 3#1)", "description": "", "templateType": "randrange", "can_override": false}, "bigint": {"name": "bigint", "group": "Ungrouped variables", "definition": "int^2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["int", "bigint", "nat", "expo", "ans2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$x=\\mathrm{log_{_{\\space\\var{nat}}}(\\var{ans2})}$</p>\n<p>[[0]]</p>\n<p></p>\n<p></p>\n<p>$\\mathrm{log}_{\\space x}\\mathrm{(\\var{bigint})=2}$</p>\n<p>[[1]]</p>\n<p></p>\n<p></p>\n<p><strong>Hint</strong>: For the following two questions, you do not need to (and cannot, anyways) work out the value of $y$ .</p>\n<p></p>\n<p></p>\n<p>$x=\\mathrm{log}_{\\space y}\\mathrm{(1)}$ , where $y$ is an unknown number.</p>\n<p>[[2]]</p>\n<p></p>\n<p></p>\n<p>$x=\\mathrm{log}_{\\space y}(y)$ , where $y$ is an unknown number.</p>\n<p>[[3]]</p>\n<p></p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "expo", "maxValue": "expo", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], 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