":"basic","front":"Why is the base restricted to $a>0$ in $a^x$?","back":"It ensures $a^x$ is real for every real $x$.","points":2,"mcqOptions":null},{"id":"d0054a0b-6d0d-4a9c-bfcb-17abdb64bd1f","cardType":"basic","front":"For $f(x)=4^x$, what is $f(-2)$?","back":"$\\frac{1}{16}$","points":1,"mcqOptions":null},{"id":"f9f64ce2-0466-4410-b70e-214f30d74b3e","cardType":"basic","front":"What is the domain of $f(x)=a^x$ for $a>0$?","back":"All real numbers.","points":1,"mcqOptions":null},{"id":"267730d9-6b32-40f5-8050-6cd6daf0156e","cardType":"basic","front":"What is the range of $f(x)=a^x$ for $a>0$?","back":"$f(x)>0$","points":1,"mcqOptions":null},{"id":"2fb462fb-fda3-47ab-ad90-b099dccf4a9e","cardType":"basic","front":"What is the $y$-intercept of $y=a^x$?","back":"$(0,1)$, since $a^0=1$.","points":1,"mcqOptions":null},{"id":"b2e68719-0637-48e0-a41d-0e01a0098ce7","cardType":"basic","front":"Why does $y=a^x$ have no $x$-intercept?","back":"Because $a^x$ is always positive.","points":2,"mcqOptions":null},{"id":"dd2e366b-827b-445d-a8a8-580d45c0faba","cardType":"basic","front":"How does $y=a^x$ behave when $a>1$?","back":"It is increasing from left to right.","points":1,"mcqOptions":null},{"id":"81e5336c-e178-4863-8906-7f9261118772","cardType":"basic","front":"For $a>1$, what happens to $a^x$ as $x\\to-\\infty$?","back":"$a^x\\to0$","points":1,"mcqOptions":null},{"id":"6174936c-276a-490d-b74e-c74f9fa999db","cardType":"basic","front":"How does $y=a^x$ behave when $0decreasing from left to right.","points":1,"mcqOptions":null},{"id":"bd86dfcd-442c-4a8a-ae02-a550015ac851","cardType":"basic","front":"For $0$a^x\\to0$","points":1,"mcqOptions":null},{"id":"79da6044-d31e-4719-9705-0dd116160c7a","cardType":"basic","front":"What is the graph of $y=1^x$?","back":"The horizontal line $y=1$.","points":1,"mcqOptions":null},{"id":"d8caa7d6-fcff-4b6a-9329-a0a5e5f27ebb","cardType":"basic","front":"What model represents an initial quantity $Q_0$ multiplied by $a$ over each of $t$ intervals?","back":"$Q=Q_0a^t$","points":1,"mcqOptions":null},{"id":"402b9e19-9f5e-4253-a61c-d711121ac60b","cardType":"basic","front":"Simplify $\\dfrac{8^{x+1}}{2^{2x}}$.","back":"$2^{x+3}$","points":2,"mcqOptions":null},{"id":"29073c97-ae6c-4293-b838-2fefe73eea4e","cardType":"basic","front":"Evaluate $\\dfrac{8^{x+1}}{2^{2x}}$ when $x=3$.","back":"$64$","points":1,"mcqOptions":null},{"id":"00449c24-460c-4b69-bbd1-6b0b20ba6665","cardType":"basic","front":"An £800 investment grows by 5% annually. What is its value after 4 years?","back":"$800(1.05)^4=£972.41$","points":2,"mcqOptions":null},{"id":"8f78a7d2-88f7-487f-8aaf-a01c423d697f","cardType":"cloze","front":"For equal positive bases, $a^m a^n=$ $\\text{[...]}$ and $\\dfrac{a^m}{a^n}=$ $a^{m-n}$.","back":"For equal positive bases, $a^m a^n=$ $\\boxed{a^{m+n}}$ and $\\dfrac{a^m}{a^n}=$ $a^{m-n}$.","points":null,"mcqOptions":null},{"id":"c9e6b8be-c7bc-4c7c-8053-54ecdbafc967","cardType":"cloze","front":"For equal positive bases, $a^m a^n=$ $a^{m+n}$ and $\\dfrac{a^m}{a^n}=$ $\\text{[...]}$.","back":"For equal positive bases, $a^m a^n=$ $a^{m+n}$ and $\\dfrac{a^m}{a^n}=$ $\\boxed{a^{m-n}}$.","points":null,"mcqOptions":null},{"id":"af50ab5a-149e-438b-b33b-dc11640356ea","cardType":"cloze","front":"A power raised to a power satisfies $(a^m)^n=$ $\\text{[...]}$, while a fractional index satisfies $a^{\\frac{1}{n}}=$ $\\sqrt[n]{a}$.","back":"A power raised to a power satisfies $(a^m)^n=$ $\\boxed{a^{mn}}$, while a fractional index satisfies $a^{\\frac{1}{n}}=$ $\\sqrt[n]{a}$.","points":null,"mcqOp

1.5.1 Exponentials and logarithms Flashcards

1.5.1 Exponentials and logarithms Flashcards