The rules for combining logs are
\n\\[\\begin{eqnarray*}&1.& \\log_b(ac)&=&\\log_b(a)+\\log_b(c)\\\\ \\\\ &2.& \\log_b\\left(\\frac{a}{c}\\right)&=&\\log_b(a)-\\log_b(c)\\\\ \\\\ &3.& \\log_b(a^r)&=&r\\log_b(a) \\end{eqnarray*} \\]
\nWe see that:
\n\\[\\begin{eqnarray*}\\log_{\\var{b}}(\\var{a_1})-\\var{r_1}\\log_{\\var{b}}(\\var{a_2})+\\var{r_2}\\log_{\\var{b}}(\\var{a_3})&=&\\log_{\\var{b}}(\\var{a_1})-\\log_{\\var{b}}(\\var{a_2}^{\\var{r_1}})+\\log_{\\var{b}}(\\var{a_3}^{\\var{r_2}})\\mbox{ using 3.}\\\\&=&\\log_{\\var{b}}(\\var{a_1})-\\log_{\\var{b}}(\\var{a_2^r_1})+\\log_{\\var{b}}(\\var{a_3^r_2})\\\\&=&\\log_{\\var{b}}(\\var{a_1}\\times \\var{a_3^r_2})-\\log_{\\var{b}}(\\var{a_2^r_1}) \\mbox{ using 1.}\\\\&=&\\log_{\\var{b}}\\left(\\frac{\\var{a_1}\\times \\var{a_3^r_2}}{\\var{a_2^r_1}}\\right) \\mbox{ using 2.}\\\\&=&\\log_{\\var{b}}\\left(\\frac{\\var{a_1*a_3^r_2}}{\\var{a_2^r_1}}\\right)\\\\&=&\\log_{\\var{b}}\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)\\mbox{ on cancelling common factors}.\\end{eqnarray*}\\]
\nHence $\\displaystyle c=\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}$.
\nTo calculate $\\displaystyle \\log_{\\var{b}}\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)$ to 4 decimal places we use the fact that for any positive base $b$:
\n\\[\\log_b(c)=\\frac{\\ln(c)}{\\ln(b)}=\\frac{\\log_{10}(c)}{\\log_{10}(b)}\\]
\nand we can use either of the log functions, $\\ln$ or $\\log_{10}$ on our calculators to find the value.
\nUsing $\\ln$ we find:
\n\\[ \\log_{\\var{b}}\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)=\\frac{\\ln\\left(\\simplify[all,fractionnumbers]{{a_1*a_3^r_2}/{a_2^r_1}}\\right)}{\\ln(\\var{b})}=\\var{ans}\\] to 4 decimal places.
\n ", "rulesets": {}, "parts": [{"prompt": "\n\n
Find a fraction or integer $c$ such that:
\n$\\log_{\\var{b}}(\\var{a_1})-\\var{r_1}\\log_{\\var{b}}(\\var{a_2})+\\var{r_2}\\log_{\\var{b}}(\\var{a_3})=\\log_{\\var{b}}(c)$
\n$c=\\;$[[0]].
\nInput $c$ as an integer or as a fraction and not as a decimal.
\nNow calculate $\\log_{\\var{b}}(c)$ to 4 decimal places:
\n$\\log_{\\var{b}}(c)=\\;$[[1]].
\n\n
\n ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "all,fractionnumbers", "marks": 2.0, "answer": "{a_1*a_3^r_2}/{a_2^r_1}", "type": "jme"}, {"minvalue": "{ans-tol}", "type": "numberentry", "maxvalue": "{ans+tol}", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "
Answer the following question on logarithms.
", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"c": {"definition": "a_1*a_3^r_2/a_2^r_1", "name": "c"}, "b": {"definition": "random(2..9)", "name": "b"}, "a_3": {"definition": "random(2..8)", "name": "a_3"}, "a_2": {"definition": "random(2,4,8)*random(3,9)", "name": "a_2"}, "a_1": {"definition": "random(5..20)", "name": "a_1"}, "r_1": {"definition": "random(2..3)", "name": "r_1"}, "r_2": {"definition": "random(2..3)", "name": "r_2"}, "tol": {"definition": 0.0001, "name": "tol"}, "ans": {"definition": "precround(log(c)/log(b),4)", "name": "ans"}}, "metadata": {"notes": "\n \t\t30/4/2013:
\n \t\t
Written for revision of logs in a foundation course
Given a sum of logs, all numbers are integers,
\n \t\t$\\log_b(a_1)+\\alpha\\log_b(a_2)+\\beta\\log_b(a_3)$ write as $\\log_b(a)$ for some fraction $a$.
\n \t\tAlso calculate to 3 decimal places $\\log_b(a)$.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}