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

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>