use quarto, not Pluto to render pages
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@@ -165,11 +165,9 @@ That would be an enormous growth. Don't worry: "Exponential growth
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cannot continue indefinitely, however, because the medium is soon
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depleted of nutrients and enriched with wastes."
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```julia; echo=false
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note("""
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The value of `2^n` and `2.0^n` is different in `Julia`. The former remains an integer and is subject to integer overflow for `n > 62`. As used above, `2^(n/6)` will not overflow for larger `n`, as when the exponent is a floating point value, the base is promoted to a floating point value.
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""")
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```
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!!! note
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The value of `2^n` and `2.0^n` is different in `Julia`. The former remains an integer and is subject to integer overflow for `n > 62`. As used above, `2^(n/6)` will not overflow for larger `n`, as when the exponent is a floating point value, the base is promoted to a floating point value.
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##### Example
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@@ -366,11 +364,9 @@ If $1/10$ of the original carbon ``14`` remains, how old is the item? This amoun
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-5730 * log2(1/10)
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```
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```julia; echo=false
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note("""
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(Historically) Libby and James Arnold proceeded to test the radiocarbon dating theory by analyzing samples with known ages. For example, two samples taken from the tombs of two Egyptian kings, Zoser and Sneferu, independently dated to ``2625`` BC plus or minus ``75`` years, were dated by radiocarbon measurement to an average of ``2800`` BC plus or minus ``250`` years. These results were published in Science in ``1949``. Within ``11`` years of their announcement, more than ``20`` radiocarbon dating laboratories had been set up worldwide. Source: [Wikipedia](http://tinyurl.com/p5msnh6).
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""")
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```
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!!! note
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(Historically) Libby and James Arnold proceeded to test the radiocarbon dating theory by analyzing samples with known ages. For example, two samples taken from the tombs of two Egyptian kings, Zoser and Sneferu, independently dated to ``2625`` BC plus or minus ``75`` years, were dated by radiocarbon measurement to an average of ``2800`` BC plus or minus ``250`` years. These results were published in Science in ``1949``. Within ``11`` years of their announcement, more than ``20`` radiocarbon dating laboratories had been set up worldwide. Source: [Wikipedia](http://tinyurl.com/p5msnh6).
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### Properties of logarithms
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@@ -503,8 +499,8 @@ Which is bigger $e^2$ or $2^e$?
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```julia; hold=true; echo=false
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choices = ["``e^2``", "``2^e``"]
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ans = e^2 - 2^e > 0 ? 1 : 2
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radioq(choices, ans)
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answ = e^2 - 2^e > 0 ? 1 : 2
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radioq(choices, answ)
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```
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@@ -514,8 +510,8 @@ Which is bigger $\log_8(9)$ or $\log_9(10)$?
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```julia; hold=true; echo=false
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choices = [raw"``\log_8(9)``", raw"``\log_9(10)``"]
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ans = log(8,9) > log(9,10) ? 1 : 2
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radioq(choices, ans)
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answ = log(8,9) > log(9,10) ? 1 : 2
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radioq(choices, answ)
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```
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###### Question
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@@ -529,8 +525,8 @@ raw"``\frac{\log(2)\log(3)}{\log(5)\log(4)}``",
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raw"``2/5``",
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raw"``\frac{\log(5)\log(4)}{\log(3)\log(2)}``"
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]
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ans = 1
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radioq(choices, ans)
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answ = 1
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radioq(choices, answ)
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```
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###### Question
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@@ -538,8 +534,8 @@ radioq(choices, ans)
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Does $12$ satisfy $\log_2(x) + \log_3(x) = \log_4(x)$?
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```julia; hold=true; echo=false
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ans = log(2,12) + log(3,12) == log(4, 12)
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yesnoq(ans)
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answ = log(2,12) + log(3,12) == log(4, 12)
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yesnoq(answ)
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```
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@@ -563,8 +559,8 @@ for each?
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```julia; hold=true; echo=false
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choices = ["``1000`` times", "``100`` times", "``10`` times", "the same"]
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ans = 2
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radioq(choices, ans, keep_order=true)
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answ = 2
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radioq(choices, answ, keep_order=true)
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```
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###### Question
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@@ -600,8 +596,8 @@ choices = [
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raw"``x \geq 1 + \log(x)``",
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raw"``x \leq 1 + \log(x)``"
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]
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ans = 1
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radioq(choices, ans)
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answ = 1
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radioq(choices, answ)
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```
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@@ -623,8 +619,8 @@ choices = [
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raw"``\log(1-x) \geq -x - x^2/2``",
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raw"``\log(1-x) \leq -x - x^2/2``"
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]
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ans = 1
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radioq(choices, ans)
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answ = 1
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radioq(choices, answ)
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```
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###### Question
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@@ -634,8 +630,8 @@ reciprocal property of exponents, $a^{-x} = (1/a)^x$, is at play here.)
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```julia; hold=true; echo=false
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choices = ["``-y``", "``1/y``", "``-1/y``"]
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ans = 1
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radioq(choices, ans)
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answ = 1
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radioq(choices, answ)
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```
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Based on this, the graph of $\log_{1/a}(x)$ is the graph of
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@@ -647,8 +643,8 @@ L"Flipped over the $x$ axis",
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L"Flipped over the $y$ axis",
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L"Flipped over the line $y=x$"
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]
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ans = 1
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radioq(choices, ans)
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answ = 1
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radioq(choices, answ)
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```
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@@ -662,8 +658,8 @@ choices = [
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raw"``a^{y-x}``",
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raw"``a^{y-x} \cdot (a^x - 1)``"
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]
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ans = 1
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radioq(choices, ans)
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answ = 1
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radioq(choices, answ)
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```
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Using $a > 1$ we have:
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@@ -674,8 +670,8 @@ choices = [
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L"as $a^x > 1$, $a^y > a^x$",
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"``a^{y-x} > 0``"
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]
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ans=1
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radioq(choices, ans)
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answ=1
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radioq(choices, answ)
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```
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If $a < 1$ then:
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@@ -686,6 +682,6 @@ L"as $a^{y-x} < 1$ as $y-x > 0$, $a^y < a^x$",
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L"as $a^x < 1$, $a^y < a^x$",
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"``a^{y-x} < 0``"
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]
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ans = 1
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radioq(choices, ans)
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answ = 1
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radioq(choices, answ)
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```
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