<p>For&nbsp;\(a&gt;0\) and \(a\ne1\):</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(\log_a1=0\qquad\log_a a=1\)</p></div></div>
<p>\(a^0=1\qquad a^1=a\)</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(\log_a(a^x)=x\qquad a^{\log_a N}=N\quad(N&gt;0)\)</p></div></div>
<p class="paragraph-header">Examples</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 53.7037%;"><col style="width: 46.2963%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Expression</strong></th><th style="text-align: center; --cell-justify: center"><strong>Value</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(\log_5 1\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(\ln e\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(\log_3(3^{-2})\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(2^{\log_2 7}\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>

\(\log_a(a^x)=x\qquad a^{\log_a N}=N\quad(N>0)\)

Exam check: \(a^{\log_a N}=N\) needs \(N>0\), whereas \(\log_a(a^x)=x\) holds for every real \(x\). For example, \(\log_2(-8)\) is undefined, so \(2^{\log_2(-8)}\) is not \(-8\).

Properties to note

Properties to note

Video Solution:

Solution:

Solution locked

Expression	Value
\(\log_5 1\)
\(\ln e\)
\(\log_3(3^{-2})\)
\(2^{\log_2 7}\)