// Numbas version: finer_feedback_settings
{"name": "Chemical Thermodynamics: Logarithm Practice - ln", "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 - ln", "tags": [], "metadata": {"description": "<p>Public</p>", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "statement": "<p>Calculate $x$ in the following expressions:</p>", "advice": "<p>You know by now to think of a logarithm as a <strong>question</strong>, written symbolically - to recap, 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>A <strong>natural logarithm</strong> is a special name given to a logarithm with base $e$, where $e=(2.718\\cdots)$ is Euler's number, an important number in this area of mathematics.</p>\n<p>Natural logarithms are also given a special symbol - instead of denoting them $\\mathrm{log}_{\\space e}$ , we denote them $\\mathrm{ln}$. You should, however, remember treat them <strong>exactly</strong> as you would treat $\\mathrm{log}_{\\space e}$ if you didn't know it was special - if you read $\\mathrm{ln}$, <strong>think</strong> $\\mathrm{log}_{\\space e}$ .</p>\n<p></p>\n<p></p>\n<p></p>\n<p>For the first problem, we must calculate the value of a natural logarithm:</p>\n<p>$x=\\mathrm{ln}(e\\mathrm{^{\\var{nat}})}$ .</p>\n<p></p>\n<p>To find this, we recall that a natural logarithm is just a <strong>logarithm of base $e$</strong>, so we must simply answer the question \"$e$ raised to which power equals $e^{\\var{nat}}$ ?\" .</p>\n<p>The answer to this question is trivial; we are already shown the power plainly - the answer to our question is $\\var{nat}$ . Thus,</p>\n<p>$x=\\mathrm{ln}(e\\mathrm{^{\\var{nat}})}=\\var{nat}$ .</p>\n<p></p>\n<p></p>\n<p></p>\n<p>For the second problem, $x$ is not the value of a natural logarithm; it is the <strong>argument</strong>&nbsp;of one:</p>\n<p>$\\mathrm{ln}(x)=\\mathrm{(\\var{expo})}$ .</p>\n<p></p>\n<p>To find $x$, let's consider the question the logarithm represents, recalling that a natural logarithm is just a <strong>logarithm of base $e$</strong>: \"$e$ raised to which power equals $x$ ?\" .</p>\n<p>As we are <strong>given the value</strong> of the logarithm, we <strong>know the answer</strong> to this question: $\\var{expo}$ .</p>\n<p>This is the \"power\" mentioned in the question, so we actually know that: \"$e$ raised to the power of $\\var{expo}$ equals $x$\" .</p>\n<p>Thus, $x=e^{\\var{expo}}=\\var{sigformat(ans2,3)}$ .</p>\n<p></p>\n<p><strong>Note</strong>: Alternatively, we can simply <strong>invert</strong> the logarithm mathematically - to invert a logarithm, for both sides of the equation, we change the expression to the logarithm's base (in this case $e$) to the power of the original expression. Thus:</p>\n<p>$\\mathrm{ln}(x)=\\mathrm{log_{\\space e}}(x)=\\mathrm{(\\var{expo})}$</p>\n<p></p>\n<p>$\\mathrm{\\Rightarrow\\space\\space\\space }x=\\mathrm{e^{\\var{expo}}=\\var{sigformat(ans2,3)}}$ .</p>\n<p></p>\n<p></p>\n<p></p>\n<p>For the third problem, we must calculate the value of a natural logarithm:</p>\n<p>$x=\\mathrm{ln}(e)$ .</p>\n<p></p>\n<p>To find this, we recall that a natural logarithm is just a <strong>logarithm of base $e$</strong>, so we must simply answer the question \"$e$ raised to which power equals $e$ ?\" .</p>\n<p>We know from earlier that the value of <strong>any number</strong> raised to the power of <strong>one</strong> is always equal to the original number, meaning that $e^1=e$ ; so, the answer to our question is simply $1$ . Thus,</p>\n<p>$x=\\mathrm{ln}(e)=1$ .</p>\n<p></p>\n<p></p>\n<p></p>\n<p>For the fourth problem, we must calculate the value of a natural logarithm:</p>\n<p>$x=\\mathrm{ln(1)}$ .</p>\n<p></p>\n<p>To find this, we recall that a natural logarithm is just a <strong>logarithm of base $e$</strong>, so we must simply answer the question \"$e$ raised to which power equals $1$ ?\" .</p>\n<p>We know from earlier that he value of <strong>any number</strong> raised to the power of <strong>zero</strong> is always equal to $1$, meaning that $e^0=1$ ; so, the answer to our question is simply $0$ . Thus,</p>\n<p>$x=\\mathrm{ln(1)=0}$ .</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"nat": {"name": "nat", "group": "Ungrouped variables", "definition": "random(2 .. 29#1)", "description": "", "templateType": "randrange", "can_override": false}, "expo": {"name": "expo", "group": "Ungrouped variables", "definition": "random(2 .. 3#1)", "description": "", "templateType": "randrange", "can_override": false}, "ans2": {"name": "ans2", "group": "Ungrouped variables", "definition": "exp(expo)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["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>$x=\\mathrm{ln}(e\\mathrm{^{\\var{nat}})}$</p>\n<p>[[0]]</p>\n<p></p>\n<p></p>\n<p>$\\mathrm{ln}(x)=\\mathrm{(\\var{expo})}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (<strong>Hint</strong>: You will probably need to use a calculator for this question).</p>\n<p>[[1]]</p>\n<p></p>\n<p></p>\n<p>$x=\\mathrm{ln}(e)$</p>\n<p>[[2]]</p>\n<p></p>\n<p></p>\n<p>$x=\\mathrm{ln(1)}$</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": "nat", "maxValue": "nat", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "ans2-(ans2/500)", "maxValue": "ans2+(ans2/500)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en", "scientific"], "correctAnswerStyle": "plain"}, {"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": "1", "maxValue": "1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Frances Docherty", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4059/"}, {"name": "Michael McFadden", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18132/"}], "resources": []}]}], "contributors": [{"name": "Frances Docherty", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4059/"}, {"name": "Michael McFadden", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18132/"}]}