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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.

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<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>