":"cloze","front":"For the same base, $a^m\\cdot a^n=$ $\\text{[...]}$.","back":"For the same base, $a^m\\cdot a^n=$ $\\boxed{a^{m+n}}$.","points":null,"mcqOptions":null},{"id":"c9417ade-265f-40b9-81ad-4e4986a8f89f","cardType":"basic","front":"Simplify $\\dfrac{a^m}{a^n}$ using index laws.","back":"$a^{m-n}$","points":1,"mcqOptions":null},{"id":"f1b8eb9b-20d3-4c25-bef4-64ba9b982ce8","cardType":"basic","front":"Simplify $\\left(a^m\\right)^n$ using index laws.","back":"$a^{mn}$","points":1,"mcqOptions":null},{"id":"3538957f-a671-430c-820a-094492871207","cardType":"cloze","front":"A negative index gives $\\text{[...]}$, while a fractional index gives $a^{\\frac{1}{n}}=\\sqrt[n]{a}$.","back":"A negative index gives $\\boxed{a^{-n}=\\dfrac{1}{a^n}}$, while a fractional index gives $a^{\\frac{1}{n}}=\\sqrt[n]{a}$.","points":null,"mcqOptions":null},{"id":"eb03c64e-32e4-4182-bb5b-b3bac401a857","cardType":"cloze","front":"A negative index gives $a^{-n}=\\dfrac{1}{a^n}$, while a fractional index gives $\\text{[...]}$.","back":"A negative index gives $a^{-n}=\\dfrac{1}{a^n}$, while a fractional index gives $\\boxed{a^{\\frac{1}{n}}=\\sqrt[n]{a}}$.","points":null,"mcqOptions":null},{"id":"3d663990-4a67-4699-9879-f48f472b57a1","cardType":"basic","front":"Evaluate $25^{-\\frac{1}{2}}$.","back":"$\\dfrac{1}{5}$","points":1,"mcqOptions":null},{"id":"22fcd54e-b8b1-4dae-924b-f34a23ed5b86","cardType":"cloze","front":"For $f(x)=a^x$, the base must satisfy $\\text{[...]}$ and $a\\neq 1$.","back":"For $f(x)=a^x$, the base must satisfy $\\boxed{a>0}$ and $a\\neq 1$.","points":null,"mcqOptions":null},{"id":"ef74ab58-2956-49bc-b2e3-00a588f646ac","cardType":"cloze","front":"For $f(x)=a^x$, the base must satisfy $a>0$ and $\\text{[...]}$.","back":"For $f(x)=a^x$, the base must satisfy $a>0$ and $\\boxed{a\\neq 1}$.","points":null,"mcqOptions":null},{"id":"40c9179b-a801-4e7d-91ec-5ec6b1afc506","cardType":"basic","front":"Why is $a=1$ excluded as the base of an exponential function with an inverse?","back":"$$1^x$ is always 1, so inputs cannot be reversed uniquely.","points":2,"mcqOptions":null},{"id":"6af9a5b4-fa4b-40f4-8d31-28653a60e198","cardType":"basic","front":"How does $y=a^x$ behave when $a>1$?","back":"It is increasing.","points":1,"mcqOptions":null},{"id":"5dce1376-04a3-4da7-b3d4-ebed27a2c861","cardType":"basic","front":"How does $y=a^x$ behave when $0decreasing.","points":1,"mcqOptions":null},{"id":"a3d1314a-91f4-4531-bba6-610c94beef36","cardType":"basic","front":"Through which fixed point does every permitted exponential graph $y=a^x$ pass?","back":"$(0,1)$","points":1,"mcqOptions":null},{"id":"aadc215c-6826-4ea7-9496-ad866396ec4e","cardType":"basic","front":"What is the range of $y=a^x$?","back":"$y>0$","points":1,"mcqOptions":null},{"id":"ffc63f0b-75de-45bc-949d-b29ba135515a","cardType":"basic","front":"Solve $3^x=81$.","back":"$x=4$","points":1,"mcqOptions":null},{"id":"29eff739-26bb-437c-8375-d13acddfbc80","cardType":"cloze","front":"Logarithmic and exponential forms are equivalent: $\\log_a x=y\\Longleftrightarrow$ $\\text{[...]}$.","back":"Logarithmic and exponential forms are equivalent: $\\log_a x=y\\Longleftrightarrow$ $\\boxed{a^y=x}$.","points":null,"mcqOptions":null},{"id":"b8494c56-8e4d-4703-a57b-0cd946c1693d","cardType":"basic","front":"Rewrite $\\log_5 125=3$ in exponential form.","back":"$5^3=125$","points":1,"mcqOptions":null},{"id":"bc9cef1c-6215-44e4-a8cd-3122a04d40a3","cardType":"basic","front":"Rewrite $2^{-3}=\\dfrac{1}{8}$ in logarithmic form.","back":"$\\log_2\\left(\\dfrac{1}{8}\\right)=-3$","points":1,"mcqOptions":null},{"id":"c0c7ea6a-59ef-42d9-acdf-45eb9d294fe3","cardType":"basic","front":"Evaluate $\\log_3\\left(\\dfrac{1}{27}\\right)$.","back":"$-3$","points":1,"mcqOptions":null},{"id":"11c1aaa9-1dff-4f6b-aed3-f11a26fed628","cardType":"basic","front":"Why is $\\log_a 0$ undefined over

1.6.4 Exponentials and logarithms Flashcards

1.6.4 Exponentials and logarithms Flashcards