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    <name>
      <text>6.3.1.4 Laws of logarithms</text>
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      <text><![CDATA[<h4>Laws of logarithms</h4><p>From the previous exercise (see feedback), we see that the index law \(a^na^m=a^{n+m}\) has an equivalent logarithm law \(\log_a(bc)=\log_ab+\log_ac\). In words: <b>"The log of a product is the sum of logs." </b></p>
<p>Indeed this property is one of the major advantages of using logarithms.They transform a product of numbers (a relatively difficult operation) to a sum of numbers (a relatively
easy operation).</p>
<p>Each index law has an equivalent logarithm law, true for any base, listed in the
following Key Point:</p>
<div class="HELM_keypoint">
<h4>Key Point</h4>
<table style="width:100%">
<thead>
<tr>
<th>The laws of logarithms</th>
<th>The laws of indices</th>
</tr>
</thead>
<tbody>
<tr><td></td><td></td></tr>
<tr>
<td>\(\log_a(AB) = \log_aA + \log_aB\)</td> <td>\(a^{A}a^B = a^{A+B} \)</td>
</tr>
<tr>
<td>\(\log_a \left( \dfrac{A}{B} \right) = \log_aA - \log_aB \)</td> <td>\(\dfrac{a^A}{a^B} = a^{A-B}\)</td>
</tr>
<tr>
<td>\(\log_a(A^k) = k\log_aA\)</td> <td>\((a^A)^k = a^{kA}\)</td>
</tr>
<tr>
<td>\(\log_a(a^A) = A\)</td> <td>\(a^{\log_aA} = A\)</td>
</tr>
<tr> 
<td> \(\log_aa = 1\)</td> <td>\(a^1 = a\)</td>
</tr>
<tr>
<td>\(\log_a1 = 0\) </td> <td> \(a^0 = 1\)</td>
</tr>
</tbody>
</table>
</div>]]></text>
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