// Numbas version: finer_feedback_settings
{"name": "Chemical Thermodynamics: Logarithm Practice - Base 10", "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 - Base 10", "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><strong><span style=\"text-decoration: underline;\">Logarithms</span></strong></p>\n<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><span style=\"text-decoration: underline;\"><strong>Question 1</strong></span></p>\n<p>For the first problem, we must calculate the value of a logarithm:</p>\n<p>$x=\\mathrm{log_{_{10}}(\\var{arg1[n1]})}$ .</p>\n<p></p>\n<p>To find this, we must simply answer the question \"$10$ raised to which power equals $\\var{arg1[n1]}$ ?\" .</p>\n<p>You will probably know that $10^{\\var{ans1}}=\\var{arg1[n1]}$ , so the answer to our question is $\\var{ans1}$ . Thus,</p>\n<p>$x=\\mathrm{log_{_{10}}(\\var{arg1[n1]})}=\\var{ans1}$ .</p>\n<p></p>\n<p></p>\n<p></p>\n<p><strong><span style=\"text-decoration: underline;\">Question 2</span></strong></p>\n<p>For the second problem, we must calculate the value of a logarithm:</p>\n<p>$x=\\mathrm{log_{_{10}}(\\var{arg2[n2]})}$ .</p>\n<p></p>\n<p>To find this, we must simply answer the question \"$10$ raised to which power equals $\\var{arg2[n2]}$ ?\" .</p>\n<p>If you are familiar with <strong>negative indices</strong>, you will probably know that $10^{\\var{ans2}}=\\frac{1}{10^{\\var{positiveans2}}}=\\var{arg2[n2]}$ , so the answer to our question is $\\var{ans2}$ . Thus,</p>\n<p>$x=\\mathrm{log_{_{10}}(\\var{arg2[n2]})}=\\var{ans2}$ .</p>\n<p></p>\n<p></p>\n<p></p>\n<p><strong>Note</strong>: If you are unfamiliar with negative indices, you just need to know this simple rule: $x^{-y}=\\frac{1}{x^y}$ , e.g. $3^{-2}=\\frac{1}{3^2}=\\frac{1}{9}$ .</p>\n<p></p>\n<p></p>\n<p><strong><span style=\"text-decoration: underline;\">Question 3</span></strong></p>\n<p>For the third problem, we must calculate the value of a logarithm:</p>\n<p>$x=\\mathrm{log_{_{10}}(\\var{arg3[n3]})}$ .</p>\n<p></p>\n<p>To find this, we must simply answer the question \"$10$ raised to which power equals $\\var{arg3[n3]}$ ?\" .</p>\n<p>The answer to this question is trivial; we are already shown the power plainly - the answer to our question is $\\var{ans3[n3]}$ . Thus,</p>\n<p>$x=\\mathrm{log_{_{10}}(\\var{arg3[n3]})}=\\var{ans3[n3]}$ .</p>\n<p></p>\n<p></p>\n<p></p>\n<p><strong><span style=\"text-decoration: underline;\">Question 4</span></strong></p>\n<p>For the fourth problem, $x$ is not the value of a logarithm; it is the <strong>argument</strong>&nbsp;of one:</p>\n<p>$\\mathrm{log_{_{10}}}(x)=\\mathrm{(\\var{int})}$ .</p>\n<p></p>\n<p>To find $x$, let's consider the question the logarithm represents: \"$10$ 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: $\\var{int}$ .</p>\n<p>This is the \"power\" mentioned in the question, so we actually know that: \"$10$ raised to the power of $\\var{int}$ equals $x$\" .</p>\n<p>Thus, $x=10^{\\var{int}}=\\var{ans4}$ .</p>\n<p></p>\n<p></p>\n<p></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 $10$) to the power of the original expression. Thus:</p>\n<p>$\\mathrm{log_{_{10}}}(x)=\\mathrm{(\\var{int})}$</p>\n<p></p>\n<p>$\\mathrm{\\Rightarrow\\space\\space\\space }x=\\mathrm{10^{\\var{int}}=\\var{ans4}}$ .</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"arg1": {"name": "arg1", "group": "Ungrouped variables", "definition": "[ 10, 100, 1000, 10000 ]", "description": "", "templateType": "list of numbers", "can_override": false}, "n1": {"name": "n1", "group": "Ungrouped variables", "definition": "random(0 .. 3#1)", "description": "", "templateType": "randrange", "can_override": false}, "arg2": {"name": "arg2", "group": "Ungrouped variables", "definition": "[ 0.1, 0.01, 0.001, 0.0001 ]", "description": "", "templateType": "list of numbers", "can_override": false}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "random(0 .. 3#1)", "description": "", "templateType": "randrange", "can_override": false}, 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