pe":"basic","front":"What makes $3^x=17$ an exponential equation?","back":"The unknown $x$ appears in the exponent.","points":1,"mcqOptions":null},{"id":"6c26dfe9-53a8-487e-af5d-7deeb28e96cd","cardType":"basic","front":"What restrictions make $a$ a valid base in $a^x=b$?","back":"$$a>0$ and $a\\neq1$.","points":1,"mcqOptions":null},{"id":"44db9b89-14d5-49dc-b5ae-4eb86e5e9638","cardType":"cloze","front":"For a valid base, $a^x$ is always [ ].","back":"For a valid base, $a^x$ is always positive.","points":null,"mcqOptions":null},{"id":"f6987c6b-bbd8-4785-90be-2f4ebe7b701c","cardType":"basic","front":"When does $a^x=b$ have no real solution?","back":"When $b\\leq0$.","points":1,"mcqOptions":null},{"id":"15673a9c-cf25-45f4-bd4c-ef2f1baad03a","cardType":"basic","front":"How many real solutions does $a^x=b$ have when $b>0$?","back":"Exactly one.","points":1,"mcqOptions":null},{"id":"696f0e24-1f19-490e-aec6-377d5e48c570","cardType":"cloze","front":"If $a^p=a^q$, where $a>0$ and $a\\neq1$, then $\\text{[...]}$.","back":"If $a^p=a^q$, where $a>0$ and $a\\neq1$, then $\\boxed{p=q}$.","points":null,"mcqOptions":null},{"id":"68837286-6c88-4751-9390-8b7adce749ba","cardType":"cloze","front":"The power-of-a-power law is $\\text{[...]}$.","back":"The power-of-a-power law is $\\boxed{\\left(a^m\\right)^n=a^{mn}}$.","points":null,"mcqOptions":null},{"id":"b5082583-bca3-4783-abe5-8bf9fb0854f2","cardType":"basic","front":"Solve $8^x=32$ by rewriting both sides with base 2.","back":"$x=\\frac{5}{3}$","points":2,"mcqOptions":null},{"id":"64542678-3667-4483-b1fd-d2adfc84d39f","cardType":"basic","front":"Solve $5^x=\\frac{1}{125}$.","back":"$x=-3$","points":2,"mcqOptions":null},{"id":"b2ae5714-612d-4e14-95a4-0ece54054523","cardType":"basic","front":"For any valid base, what is $x$ if $a^x=1$?","back":"$x=0$","points":1,"mcqOptions":null},{"id":"35a35e44-7460-461c-88be-8a2268fca450","cardType":"cloze","front":"The logarithmic form of $a^x=b$ is $\\text{[...]}$.","back":"The logarithmic form of $a^x=b$ is $\\boxed{x=\\log_a b}$.","points":null,"mcqOptions":null},{"id":"905b890a-8327-4d4b-9965-6712c0c48b40","cardType":"basic","front":"Why are logarithms used when solving equations such as $3^x=17$?","back":"They give the exponent needed to produce 17.","points":2,"mcqOptions":null},{"id":"262f9b83-e913-4b36-9ea5-d74d1e0fbacb","cardType":"cloze","front":"The change-of-base formula using common logarithms is $\\text{[...]}$.","back":"The change-of-base formula using common logarithms is $\\boxed{\\log_a b=\\frac{\\log b}{\\log a}}$.","points":null,"mcqOptions":null},{"id":"d93e7992-211a-4929-9afc-7d37eb521977","cardType":"cloze","front":"The change-of-base formula using natural logarithms is $\\text{[...]}$.","back":"The change-of-base formula using natural logarithms is $\\boxed{\\log_a b=\\frac{\\ln b}{\\ln a}}$.","points":null,"mcqOptions":null},{"id":"0fb2e619-2f15-4895-b9e0-7582d75eb08d","cardType":"basic","front":"After taking logarithms of $a^x=b$, which law brings $x$ down from the exponent?","back":"$\\log(a^x)=x\\log a$","points":1,"mcqOptions":null},{"id":"0a150784-ea94-4e3b-b1c7-c2926605845a","cardType":"basic","front":"What exact solution follows from $x\\log a=\\log b$?","back":"$x=\\frac{\\log b}{\\log a}$","points":1,"mcqOptions":null},{"id":"374e7c13-d55c-419d-8597-c3fb602ee3a3","cardType":"basic","front":"How should $x=\\frac{\\ln b}{\\ln a}$ be presented when an exact answer is required?","back":"Leave it in logarithmic form.","points":1,"mcqOptions":null},{"id":"53e7bd59-616b-4df7-b44e-6307db1c8d33","cardType":"basic","front":"Why should separate logarithms not be rounded before division?","back":"Early rounding causes inaccuracy in the final answer.","points":2,"mcqOptions":null},{"id":"444f87ed-6959-4be6-b444-c872255a9f77","cardType":"b

1.9.5 Solving exponential equations Flashcards

1.9.5 Solving exponential equations Flashcards