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{"name": "Jamie's copy of Graphing logarithms of the form y=log_{b}(x) with b>1", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>The following questions will gauge your understanding of logarithms and how to graph them.&nbsp;</p>\n<p>The logarithm you will be working with for this question is \\[y=\\log_{\\var{b}}(x).\\]</p>\n<p></p>", "variables": {"b": {"group": "Ungrouped variables", "name": "b", "templateType": "anything", "description": "", "definition": "random(2..10)"}}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>The easiest type of exponential to <g class=\"gr_ gr_3 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar only-ins doubleReplace replaceWithoutSep\" id=\"3\" data-gr-id=\"3\">graph</g> where the base is greater than 1 and no transformations take place.</p>"}, "parts": [{"variableReplacements": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "prompt": "<p>The $x$-intercept of $y=\\log_{\\var{b}}(x)$ is the point $\\large($[[0]], 0$\\large)$.</p>", "showCorrectAnswer": true, "gaps": [{"correctAnswerStyle": "plain", "marks": 1, "variableReplacementStrategy": "originalfirst", "maxValue": "1", "variableReplacements": [], "showCorrectAnswer": true, "minValue": "1", "type": "numberentry", "correctAnswerFraction": false, "showFeedbackIcon": true, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "scripts": {}, "allowFractions": false}], "type": "gapfill"}, {"variableReplacements": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "prompt": "<p>Another easily found point on the curve is $\\large($[[0]], $\\large1 )$.</p>", "showCorrectAnswer": true, "gaps": [{"correctAnswerStyle": "plain", "marks": 1, "variableReplacementStrategy": "originalfirst", "maxValue": "{b}", "variableReplacements": [], "showCorrectAnswer": true, "minValue": "{b}", "type": "numberentry", "correctAnswerFraction": true, "showFeedbackIcon": true, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "scripts": {}, "allowFractions": true}], "type": "gapfill"}, {"variableReplacements": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "prompt": "<p>Given $y=\\log_{\\var{b}}(x)$, everytime&nbsp;$x$ increases by a factor of $\\var{b}$, $y$ &nbsp;[[0]].</p>", "showCorrectAnswer": true, "gaps": [{"shuffleChoices": true, "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "type": "1_n_2", "matrix": ["1", 0, 0, 0, "0", 0], "showFeedbackIcon": true, "maxMarks": 0, "displayColumns": 0, "scripts": {}, "distractors": ["", "", "", "", "", ""], "displayType": "dropdownlist", "minMarks": 0, "choices": ["<p>increases by 1.</p>", "<p>decreases by 1.</p>", "<p>increases by {b-1}.</p>", "<p>decreases by {b-1}.</p>", "<p>increases by a factor of {b}.</p>", "<p>decreases by a factor of {b}.</p>"]}], "type": "gapfill"}, {"variableReplacements": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "prompt": "<p>As $x$ increases without bound, [[0]]</p>", "showCorrectAnswer": true, "gaps": [{"shuffleChoices": true, "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "type": "1_n_2", "matrix": [0, 0, "1", 0], "showFeedbackIcon": true, "maxMarks": 0, "displayColumns": "1", "scripts": {}, "distractors": ["", "", "", ""], "displayType": "radiogroup", "minMarks": 0, "choices": ["<p>$y$ approaches $0$.</p>", "<p>$y$ decreases without bound.</p>", "<p>$y$ increases without bound.</p>", "<p>$y$ approaches $\\var{b}$.</p>"]}], "type": "gapfill"}, {"shuffleChoices": true, "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "type": "1_n_2", "matrix": ["1", 0, 0, 0], "showFeedbackIcon": true, "maxMarks": 0, "displayColumns": "2", "prompt": "<p>Which graph best represents $y=\\log_{\\var{b}}(x)$?</p>", "scripts": {}, "distractors": ["", "", "", ""], "displayType": "radiogroup", "minMarks": 0, "choices": ["<p>{graph1(1)}</p>", "<p>{graph1(2)}</p>", "<p>{graph1(3)}</p>", "<p>{graph1(4)}</p>"]}], "tags": [], "ungrouped_variables": ["b"], "functions": {"graph1": {"parameters": [["quad", "number"]], "definition": "var div = Numbas.extensions.jsxgraph.makeBoard('300px','300px',{boundingBox:[-12,12,12,-12],grid:true,axis:false});\nvar board = div.board;\n\n// create the x-axis.\nvar xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: 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"<p>This question assumes you understand the definition and the laws of logarithms.</p>\n<p>a) To find the $x$-intercept, substitute $y=0$ into the equation: $0=\\log_{\\var{b}}(x)$ which is equivalent to $\\var{b}^0=x$, i.e. $x=1$. Therefore,&nbsp;the $x$-intercept is the point $(1,0)$.</p>\n<p>b) Substitute $y=1$ into the equation:&nbsp;$1=\\log_{\\var{b}}(x)$ which is equivalent to $\\var{b}^1=x$, i.e. $x=\\var{b}$. Therefore, another easily found point is $(\\var{b},1)$.</p>\n<p>c) Let's investigate what happens to the value of $y$ when the value of $x$ is increased by a factor of $\\var{b}$:</p>\n<p>\\[\\log_{\\var{b}}(\\var{b}x)=\\log_{\\var{b}}(\\var{b})+\\log_{\\var{b}}(x)=1+\\log_{\\var{b}}(x)\\]</p>\n<p>This is the old $y$ value plus 1, so we can say that $y$ is increased by 1.</p>\n<p>d)&nbsp;As $x$ gets larger and larger, $y$ gets larger and larger. Even though the rate of increase slows, $y$ continues to grow without bound.</p>\n<p>e) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve $y=\\log_{\\var{b}}(x)$ the closer $x$ gets to zero (approaching from the right), the smaller $y$ gets (without bound). In other words, as $x$ approaches $0$ from the right, $y$ approaches negative infinity. This means that the asymptote for $y=\\log_{\\var{b}}(x)$ is the line $x=0$ (the $y$-axis).</p>\n<p>f) Given all the information above, it should be clear that the graph should look like&nbsp;</p>\n<p>{graph1(1)}</p>", "name": "Jamie's copy of Graphing logarithms of the form y=log_{b}(x) with b>1", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "contributors": [{"name": "Jamie Wood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/79/"}]}]}], "contributors": [{"name": "Jamie Wood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/79/"}]}