2a:null 2b:["$","title",null,{"children":"The logarithm function | Mattias Villani | Observable"}] 31:T655,To define the logarithm, let us start with some concrete examples. - The **logarithm with base 10** (the 10-logarithm) of the number ${tex`1000`} is 3, because 1000 is 10 to the 3rd power: ${tex.block`1000 = 10^3.`} We write the 10-logarithm as ${tex`\log_{10}`}. - The **logarithm with base 2** (the 2-logarithm) of the number ${tex`256`} is 8, because 256 is 2 to the 8th power ${tex.block`256 = 2^8.`} We write the 2-logarithm as ${tex`\log_{2}`}. - The **natural logarithm** of the number ${tex`256`} is approximately 5.5451774, because ${tex.block`e^{5.5451774} \approx 256.`} where ${tex`e\approx 2.71828`} is Euler's number, which is therefore the base for the natural logarithm. We write the natural logarithm as ${tex`\log_{e}`}, or ${tex`\ln`} or simply ${tex`\log`}. Generally, the logarithm with base *b* of the number ${tex`b^a`} is *a*. The [logarithm function](https://en.wikipedia.org/wiki/Logarithm) ${tex`\log_b(x)`} with base ${tex`b`} is the [inverse function](https://en.wikipedia.org/wiki/Inverse_function) of the exponential function ${tex`b^x`} with base *b*. This means that for any *x* ${tex.block`\log_b(b^x) = x`} In particular, the natural logarithm is the inverse of the [natural exponential function](https://observablehq.com/@mattiasvillani/the-exponential-function) ${tex.block`\ln e^x = x`} Logarithms have several important properties (see below), including that they - turn products into sums: ${tex`\log_b(x \cdot y) = \log_b(x) + \log_b(y)`} - 'pull down exponents': ${tex`\log_b(x^y) = y\log_b(x)`} Someone on the internet (source lost) expressed the second property as:32:T62c,Plot.plot({ title: "The logarithm is the inverse of the exponential function", style: "font-size: 12px;", aspectRatio: 1, color: { legend: true, domain: ["linear", "exponential base b", "logarithm base b"], range: [colors[1],colors[0], colors[2]] }, x: { label: "x", domain: [-11,11] }, y: { label: "f(x)", domain: [-11,11], }, marks: [ Plot.ruleY([0]), Plot.ruleX([0]), Plot.line([{x: b, y: 0},{x: b, y:1}], {filter: plotbase, x: "x", y: "y", stroke: "gray", strokeDasharray: "4,4"}), Plot.line([{x: 0, y: 1},{x: b, y:1}], {filter: plotbase, x: "x", y: "y", stroke: "gray", strokeDasharray: "4,4"}), Plot.text([{x: b, y: 0}], {filter: plotbase, x: "x", y: "y", text: [`b = ${b.toFixed(3)}`], textAnchor: "start", fontSize: 16, dx: -3, dy: 22, fill: "gray"}), Plot.lineY(explogdata, {x: "x", y: "lin", stroke: colors[1], strokeDasharray: '5,5'}), Plot.lineY(explogdata, {x: "x", y: "exp", stroke: colors[0], strokeWidth: 2.5}), Plot.lineY(explogdata, {x: "x", y: "log", stroke: colors[2], strokeWidth: 2.5}), Plot.dot([{x: 1, y: 0},{x: 0, y: 1}], {x: "x", y: "y", stroke: [colors[2],colors[0]], fill: [colors[2],colors[0]], strokeWidth: 2}), Plot.text([{x: 1, y: 0}], {x: "x", y: "y", dy: 15, dx: 10, fontSize: 16, text: ["(1,0)"]}), Plot.text([{x: 0, y: 1}], {x: "x", y: "y", dy: -10, dx: -15, fontSize: 16, text: ["(0,1)"]}), ] })33:[{"id":"887e44157ad5d82621eb3237bf1b25b112f8c136a777db99c1b602cfbad2335f0a5952767c1780ab3382cef00f88aeda69459bcc64cc80864303123599c8196e","href":"https://static.observableusercontent.com/files/887e44157ad5d82621eb3237bf1b25b112f8c136a777db99c1b602cfbad2335f0a5952767c1780ab3382cef00f88aeda69459bcc64cc80864303123599c8196e","name":"emojilog.png","type":"image/png","lastModified":1741872494330.983,"size":12729}] 2e:{"id":"2a7b465c846f0127","slug":"the-logarithm-function","title":"The logarithm function","thumbnail":"5e62e52b35d7938c496e6e7ef587a30f0630175ea958809a140710c114246702","access_level":"public","create_time":"$D2025-03-05T14:42:02.934Z","update_time":"$D2025-03-20T19:33:41.801Z","view_time":null,"first_public_time":"$D2025-03-05T14:42:02.934Z","fork":{"id":"b3fedb483c609af0","slug":"the-exponential-function","title":"The exponential function","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias 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\"base, b\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"Plot.plot({\n title: \"The logarithm function\",\n style: \"font-size: 12px;\",\n x: {\n label: \"x\",\n domain: [-2,5] \n },\n y: {\n label: \"f(x) = b^x\",\n domain: [-11,7],\n },\n marks: [\n Plot.line([{x: b, y: 0},{x: b, y:1}], \n {filter: plotbase, x: \"x\", y: \"y\", stroke: \"gray\", strokeDasharray: \"4,4\"}),\n Plot.line([{x: 0, y: 1},{x: b, y:1}], \n {filter: plotbase, x: \"x\", y: \"y\", stroke: \"gray\", strokeDasharray: \"4,4\"}),\n Plot.text([{x: b, y: 0}], \n {filter: plotbase, x: \"x\", y: \"y\", text: [`b = ${b.toFixed(3)}`], textAnchor: \"start\", fontSize: 16, \n dx: -3, dy: 22, fill: \"gray\"}), \n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(funcdata, {x: \"x\", y: \"y\", stroke: colors[2], strokeWidth: 3})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":209,"value":"propertiestext = html`

\n \n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \n \n\n \n \n \n\n

Properties of the natural logarithm

${tex`\\log(e) = 1`}

${tex`\\log(1) = 0`}

${tex`\\log(x \\cdot y) = \\log x + \\log y`}

${tex`\\log(\\frac{x}{y}) = \\log x - \\log y`}

${tex`\\log x^y = y \\log x`}

${tex`\\log e^y = y \\log e = y`}

Inverse: ${tex`e^x`} so that ${tex`\\log(e^x) = x`}

Derivative: ${tex`\\frac{\\mathrm{d}}{\\mathrm{d}x}\\log(x) = \\frac{1}{x}`}

Integral: ${tex`\\int \\log(x) = x \\log(x) - x + C`}

\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":232,"value":"Most of these properties hold for exponential functions ${tex`b^x`} in some other base ${tex`b`}.
\nFor example, ${tex`\\log_b(b)=1`} and ${tex`\\log_b(x \\cdot y) = \\log_b(x) + \\log_b(y)`}.\n\nThe graph below illustrates that the logarithm and exponential functions are each other's inverse functions.","pinned":false,"hidden":false,"mode":"md"},{"id":311,"value":"Inputs.bind(Inputs.range([2, 10], {bind: true, value: 2.71, step: 0.01, label: \"base, b\"}), viewof b)","pinned":false,"hidden":false,"mode":"ojs"},{"id":365,"value":"viewof plotbase = Inputs.toggle({label: \"plot base\", value: false})","pinned":false,"hidden":false,"mode":"ojs"},{"id":264,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":14,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is some code and data details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":16,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":18,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":true,"hidden":false,"mode":"ojs"},{"id":20,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs"},{"id":23,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":25,"value":"function f(x, b){\n return mathjs.log(x, b)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":40,"value":"xdomain = [0.01,3]","pinned":false,"hidden":false,"mode":"ojs"},{"id":36,"value":"funcdata = {\n return d3.range(-11, 11, 22/1000).map(d => ({x: d, y: f(d, b)}))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":260,"value":"explogdata = {\n return d3.range(-11, 11, 2*(xdomain[1])/1000).map(d => ({x: d, lin: d, log: Math.log(d)/Math.log(b), exp: b**d}))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":227,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":229,"value":"","pinned":false,"hidden":false,"mode":"html"}],"files":"$W33","stdlib":"1","titling":"auto"}} 2f:"$2e:body"