admin2/openfreebooks-bookgenerator
1
0
Fork 0
mirror of https://github.com/HananoshikaYomaru/openfreebooks.git synced 2026-11-13 13:05:51 +00:00
Code Issues Projects Releases Packages Wiki Activity Actions

Enhance chapter content and metadata: Update SKILL.md to include requirements for chart axes in all widgets. Implement axis labels, arrowheads, and graduations in the simultaneous graph explorer widget. Update math curriculum JSON to reflect the status change of multiple chapters to live. Add last modified timestamps to chapter templates and generate corresponding metadata for improved site navigation and SEO.

Browse source
This commit is contained in:
Hananoshika Yomaru 2026-05-21 00:33:01 -07:00
parent 353b129533
commit eee5d109d5
31 changed files with 3093 additions and 17 deletions
Download patch file Download diff file Expand all files Collapse all files

262
content/math/exponential-logarithmic/_index.md Normal file
View file

@ -0,0 +1,262 @@
+++
title = "Exponential & logarithmic functions"
description = "Use exponential and logarithmic models, apply log laws, and solve growth and decay equations."
template = "chapter.html"
weight = 16
[extra]
subject = "math"
chapter_id = "exponential-logarithmic"
strand = "Number & Algebra"
+++
<div class="book-prose">
<p>
Exponential and logarithmic functions are inverse ideas. They model fast growth, rapid decay,
and scale changes that linear models cannot describe well.
</p>
<h2 id="exponential-basics" class="book-prose__heading">Exponential basics</h2>
<p>
An exponential function has variable in the exponent:
\[
y=a^x,\quad a&gt;0,\ a\ne1.
\]
</p>
<p>
If \(a&gt;1\), the function grows as \(x\) increases. If \(0&lt;a&lt;1\), it decays.
</p>
<div class="book-callout book-callout--formula" role="note">
<p class="book-callout__label">Example</p>
<p class="book-formula">\[
2^x \text{ grows},\qquad \left(\frac12\right)^x \text{ decays}.
\]</p>
</div>
<h2 id="logarithm-meaning" class="book-prose__heading">What is a logarithm?</h2>
<p>
A logarithm asks for the exponent:
\[
\log_a b = c \iff a^c=b.
\]
</p>
<p>
So \(\log_2 8=3\) because \(2^3=8\). Also, \(\log_{10}100=2\).
</p>
<h2 id="domain-and-range" class="book-prose__heading">Domain and range</h2>
<ul>
<li>\(y=a^x\): domain all real \(x\), range \(y&gt;0\), horizontal asymptote \(y=0\).</li>
<li>\(y=\log_a x\): domain \(x&gt;0\), range all real numbers, vertical asymptote \(x=0\).</li>
</ul>
<p>
These two graphs are reflections of each other in the line \(y=x\).
</p>
<h2 id="laws-indices" class="book-prose__heading">Index laws review</h2>
<p>
Exponential solving often uses:
</p>
<p class="book-formula">
\[
a^m a^n=a^{m+n},\quad
\frac{a^m}{a^n}=a^{m-n},\quad
(a^m)^n=a^{mn}.
\]
</p>
<h2 id="laws-logs" class="book-prose__heading">Logarithm laws</h2>
<p>
For \(M,N&gt;0\):
</p>
<p class="book-formula">
\[
\log_a(MN)=\log_a M+\log_a N,
\]
\[
\log_a\!\left(\frac{M}{N}\right)=\log_a M-\log_a N,
\]
\[
\log_a(M^k)=k\log_a M.
\]
</p>
<p><strong>Common mistakes:</strong> \(\log(M+N)\ne \log M+\log N\).</p>
<h2 id="change-of-base" class="book-prose__heading">Change of base</h2>
<p>
To evaluate logs with calculator:
\[
\log_a b = \frac{\log b}{\log a} = \frac{\ln b}{\ln a}.
\]
</p>
<p>
Here \(\log\) usually means base 10 and \(\ln\) means base \(e\).
</p>
<h2 id="solve-exponential" class="book-prose__heading">Solving exponential equations</h2>
<p><strong>Case 1: same base.</strong></p>
<p class="book-formula">\[
3^{2x-1}=3^5 \Rightarrow 2x-1=5 \Rightarrow x=3.
\]</p>
<p><strong>Case 2: take logs.</strong></p>
<p class="book-formula">\[
5^x=17 \Rightarrow x=\log_5 17=\frac{\log 17}{\log 5}\approx 1.760.
\]</p>
<h2 id="solve-log" class="book-prose__heading">Solving logarithmic equations</h2>
<p><strong>Example 1.</strong> \(\log_2(x-1)=4\)</p>
<p>
Convert to exponential: \(x-1=2^4=16\), so \(x=17\). Domain check: \(x&gt;1\), valid.
</p>
<p><strong>Example 2.</strong> \(\log_{10}(x+3)+\log_{10}(x-1)=1\)</p>
<p>
\[
\log_{10}\big((x+3)(x-1)\big)=1
\Rightarrow (x+3)(x-1)=10.
\]
\[
x^2+2x-3=10 \Rightarrow x^2+2x-13=0.
\]
\[
x=-1\pm\sqrt{14}.
\]
Domain requires \(x&gt;1\), so \(x=-1+\sqrt{14}\) only.
</p>
<h2 id="growth-decay-models" class="book-prose__heading">Growth and decay models</h2>
<p>
A common model:
\[
A(t)=A_0(1+r)^t
\]
for growth (\(r&gt;0\)) or decay (\(r&lt;0\), with \(1+r&gt;0\)).
</p>
<p>
Continuous model:
\[
A(t)=A_0e^{kt},
\]
where \(k&gt;0\) gives growth and \(k&lt;0\) gives decay.
</p>
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
<ul>
<li>State domain restrictions first in log equations.</li>
<li>When combining logs, check arguments stay positive.</li>
<li>Use exact forms before calculator rounding.</li>
<li>For modeling, define variables and units clearly.</li>
</ul>
</div>
<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
Logarithms were developed in the early 17th century (notably by John Napier) to simplify long
multiplication and division into addition and subtraction. Before electronic calculators, this was
a major speed boost for astronomy, navigation, and engineering.
</p>
<p>
Exponential functions became central in science through models of population growth, radioactive
decay, and continuously changing quantities.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
<h3 id="why-log-laws" class="book-prose__heading">Why log product law works</h3>
<p>
Let \(\log_a M=p\) and \(\log_a N=q\). Then \(M=a^p\), \(N=a^q\), so
\[
MN=a^{p+q}.
\]
Taking \(\log_a\) gives:
\[
\log_a(MN)=p+q=\log_a M+\log_a N.
\]
</p>
<h3 id="inverse-view" class="book-prose__heading">Inverse relationship view</h3>
<p>
If \(y=a^x\), swapping \(x,y\) gives \(x=a^y\), so \(y=\log_a x\). That is why exponential and
logarithmic graphs are mirror images across \(y=x\).
</p>
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
<details class="book-question">
<summary class="book-question__prompt">
Evaluate \(\log_3 81\) and \(\log_{10}0.01\).
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(\log_3 81=4\), \(\log_{10}0.01=-2\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
Expand: \(\log_2(8x^3)\).
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(\log_2 8 + \log_2 x^3 = 3 + 3\log_2 x\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
Solve \(2^{x+1}=7\).
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(x+1=\log_2 7\), so \(x=\log_2 7-1 \approx 1.807\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
Solve \(\log_5(x-4)=2\) and state domain condition.
</summary>
<div class="book-question__solution">
<p><strong>Domain:</strong> \(x-4&gt;0\Rightarrow x&gt;4\).</p>
<p><strong>Answer:</strong> \(x-4=25\Rightarrow x=29\), valid.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
A population starts at 500 and grows 6% per year. Write the model and find \(A(4)\).
</summary>
<div class="book-question__solution">
<p><strong>Model:</strong> \(A(t)=500(1.06)^t\).</p>
<p><strong>Value:</strong> \(A(4)=500(1.06)^4\approx 631.24\), about 631.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
A substance follows \(A(t)=120e^{-0.18t}\). Is this growth or decay? Why?
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Decay, because exponent coefficient is negative (\(k=-0.18\)).</p>
</div>
</details>
<h2 id="applications" class="book-prose__heading">Applications</h2>
<h3 id="compound-interest" class="book-prose__heading">Compound interest</h3>
<p>
Savings and loans often follow exponential rules. Logarithms are used to solve for unknown time
or interest rate.
</p>
<h3 id="half-life" class="book-prose__heading">Half-life and decay</h3>
<p>
Radioactive substances and medicine concentration in the body are modeled with exponential decay.
</p>
<h3 id="scales" class="book-prose__heading">Logarithmic scales</h3>
<p>
pH, decibels, and Richter magnitude use logarithmic scales to represent very large ratio ranges
in manageable numbers.
</p>
<h2 id="references" class="book-prose__heading">References</h2>
<ul>
<li>Standard high-school curricula on exponential and logarithmic functions (IGCSE, IB, A-Level, AP).</li>
<li>Precalculus and algebra references on log laws, inverse functions, and modeling.</li>
</ul>
</div>

138
content/math/exponential-logarithmic/core.html Normal file
View file

@ -0,0 +1,138 @@
<div class="book-prose">
<p>
Exponential and logarithmic functions are inverse ideas. They model fast growth, rapid decay,
and scale changes that linear models cannot describe well.
</p>
<h2 id="exponential-basics" class="book-prose__heading">Exponential basics</h2>
<p>
An exponential function has variable in the exponent:
\[
y=a^x,\quad a&gt;0,\ a\ne1.
\]
</p>
<p>
If \(a&gt;1\), the function grows as \(x\) increases. If \(0&lt;a&lt;1\), it decays.
</p>
<div class="book-callout book-callout--formula" role="note">
<p class="book-callout__label">Example</p>
<p class="book-formula">\[
2^x \text{ grows},\qquad \left(\frac12\right)^x \text{ decays}.
\]</p>
</div>
<h2 id="logarithm-meaning" class="book-prose__heading">What is a logarithm?</h2>
<p>
A logarithm asks for the exponent:
\[
\log_a b = c \iff a^c=b.
\]
</p>
<p>
So \(\log_2 8=3\) because \(2^3=8\). Also, \(\log_{10}100=2\).
</p>
<h2 id="domain-and-range" class="book-prose__heading">Domain and range</h2>
<ul>
<li>\(y=a^x\): domain all real \(x\), range \(y&gt;0\), horizontal asymptote \(y=0\).</li>
<li>\(y=\log_a x\): domain \(x&gt;0\), range all real numbers, vertical asymptote \(x=0\).</li>
</ul>
<p>
These two graphs are reflections of each other in the line \(y=x\).
</p>
<h2 id="laws-indices" class="book-prose__heading">Index laws review</h2>
<p>
Exponential solving often uses:
</p>
<p class="book-formula">
\[
a^m a^n=a^{m+n},\quad
\frac{a^m}{a^n}=a^{m-n},\quad
(a^m)^n=a^{mn}.
\]
</p>
<h2 id="laws-logs" class="book-prose__heading">Logarithm laws</h2>
<p>
For \(M,N&gt;0\):
</p>
<p class="book-formula">
\[
\log_a(MN)=\log_a M+\log_a N,
\]
\[
\log_a\!\left(\frac{M}{N}\right)=\log_a M-\log_a N,
\]
\[
\log_a(M^k)=k\log_a M.
\]
</p>
<p><strong>Common mistakes:</strong> \(\log(M+N)\ne \log M+\log N\).</p>
<h2 id="change-of-base" class="book-prose__heading">Change of base</h2>
<p>
To evaluate logs with calculator:
\[
\log_a b = \frac{\log b}{\log a} = \frac{\ln b}{\ln a}.
\]
</p>
<p>
Here \(\log\) usually means base 10 and \(\ln\) means base \(e\).
</p>
<h2 id="solve-exponential" class="book-prose__heading">Solving exponential equations</h2>
<p><strong>Case 1: same base.</strong></p>
<p class="book-formula">\[
3^{2x-1}=3^5 \Rightarrow 2x-1=5 \Rightarrow x=3.
\]</p>
<p><strong>Case 2: take logs.</strong></p>
<p class="book-formula">\[
5^x=17 \Rightarrow x=\log_5 17=\frac{\log 17}{\log 5}\approx 1.760.
\]</p>
<h2 id="solve-log" class="book-prose__heading">Solving logarithmic equations</h2>
<p><strong>Example 1.</strong> \(\log_2(x-1)=4\)</p>
<p>
Convert to exponential: \(x-1=2^4=16\), so \(x=17\). Domain check: \(x&gt;1\), valid.
</p>
<p><strong>Example 2.</strong> \(\log_{10}(x+3)+\log_{10}(x-1)=1\)</p>
<p>
\[
\log_{10}\big((x+3)(x-1)\big)=1
\Rightarrow (x+3)(x-1)=10.
\]
\[
x^2+2x-3=10 \Rightarrow x^2+2x-13=0.
\]
\[
x=-1\pm\sqrt{14}.
\]
Domain requires \(x&gt;1\), so \(x=-1+\sqrt{14}\) only.
</p>
<h2 id="growth-decay-models" class="book-prose__heading">Growth and decay models</h2>
<p>
A common model:
\[
A(t)=A_0(1+r)^t
\]
for growth (\(r&gt;0\)) or decay (\(r&lt;0\), with \(1+r&gt;0\)).
</p>
<p>
Continuous model:
\[
A(t)=A_0e^{kt},
\]
where \(k&gt;0\) gives growth and \(k&lt;0\) gives decay.
</p>
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
<ul>
<li>State domain restrictions first in log equations.</li>
<li>When combining logs, check arguments stay positive.</li>
<li>Use exact forms before calculator rounding.</li>
<li>For modeling, define variables and units clearly.</li>
</ul>
</div>

112
content/math/exponential-logarithmic/supplement.html Normal file
View file

@ -0,0 +1,112 @@
<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
Logarithms were developed in the early 17th century (notably by John Napier) to simplify long
multiplication and division into addition and subtraction. Before electronic calculators, this was
a major speed boost for astronomy, navigation, and engineering.
</p>
<p>
Exponential functions became central in science through models of population growth, radioactive
decay, and continuously changing quantities.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
<h3 id="why-log-laws" class="book-prose__heading">Why log product law works</h3>
<p>
Let \(\log_a M=p\) and \(\log_a N=q\). Then \(M=a^p\), \(N=a^q\), so
\[
MN=a^{p+q}.
\]
Taking \(\log_a\) gives:
\[
\log_a(MN)=p+q=\log_a M+\log_a N.
\]
</p>
<h3 id="inverse-view" class="book-prose__heading">Inverse relationship view</h3>
<p>
If \(y=a^x\), swapping \(x,y\) gives \(x=a^y\), so \(y=\log_a x\). That is why exponential and
logarithmic graphs are mirror images across \(y=x\).
</p>
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
<details class="book-question">
<summary class="book-question__prompt">
Evaluate \(\log_3 81\) and \(\log_{10}0.01\).
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(\log_3 81=4\), \(\log_{10}0.01=-2\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
Expand: \(\log_2(8x^3)\).
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(\log_2 8 + \log_2 x^3 = 3 + 3\log_2 x\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
Solve \(2^{x+1}=7\).
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(x+1=\log_2 7\), so \(x=\log_2 7-1 \approx 1.807\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
Solve \(\log_5(x-4)=2\) and state domain condition.
</summary>
<div class="book-question__solution">
<p><strong>Domain:</strong> \(x-4&gt;0\Rightarrow x&gt;4\).</p>
<p><strong>Answer:</strong> \(x-4=25\Rightarrow x=29\), valid.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
A population starts at 500 and grows 6% per year. Write the model and find \(A(4)\).
</summary>
<div class="book-question__solution">
<p><strong>Model:</strong> \(A(t)=500(1.06)^t\).</p>
<p><strong>Value:</strong> \(A(4)=500(1.06)^4\approx 631.24\), about 631.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">
A substance follows \(A(t)=120e^{-0.18t}\). Is this growth or decay? Why?
</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Decay, because exponent coefficient is negative (\(k=-0.18\)).</p>
</div>
</details>
<h2 id="applications" class="book-prose__heading">Applications</h2>
<h3 id="compound-interest" class="book-prose__heading">Compound interest</h3>
<p>
Savings and loans often follow exponential rules. Logarithms are used to solve for unknown time
or interest rate.
</p>
<h3 id="half-life" class="book-prose__heading">Half-life and decay</h3>
<p>
Radioactive substances and medicine concentration in the body are modeled with exponential decay.
</p>
<h3 id="scales" class="book-prose__heading">Logarithmic scales</h3>
<p>
pH, decibels, and Richter magnitude use logarithmic scales to represent very large ratio ranges
in manageable numbers.
</p>
<h2 id="references" class="book-prose__heading">References</h2>
<ul>
<li>Standard high-school curricula on exponential and logarithmic functions (IGCSE, IB, A-Level, AP).</li>
<li>Precalculus and algebra references on log laws, inverse functions, and modeling.</li>
</ul>
</div>