openfreebooks-bookgenerator/content/math/exponential-logarithmic/supplement.html

<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>