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<!--replace-end-7--><!--replace-end-4--><!--replace-end-1--></head><body><div class="ui fluid container universe"><!--replace-start-2--><!--replace-start-3--><!--replace-start-6--><div class="ui text container" id="zettel-container" style="position: relative"><div class="zettel-view"><article class="ui raised attached segment zettel-content"><div class="pandoc"><h1 id="title-h1">Logarithms</h1><p>Most simply a logarithm is a way of answering the question:</p><blockquote><p>How many of one number do we need to get another number. How many of x do we need to get y</p></blockquote><p>More formally:</p><blockquote><p>x raised to what power gives me y</p></blockquote><p>Below is an example of a logarithm:</p><p>$$ \log _{3} 9</p><p>$$</p><p>We read it:</p><blockquote><p>log base 3 of 9</p></blockquote><p>And it means:</p><blockquote><p>3 raised to what power gives me 9?</p></blockquote><p>In this case the answer is easy: <span class="math inline">\(3^2\)</span> gives me nine, which is to say: three multiplied by itself.</p><h2 id="using-exponents-to-calculate-logarithms">Using exponents to calculate logarithms</h2><p>This approach becomes rapidly difficult when working with larger numbers. It’s not as obvious what <span class="math inline">\(\log \_{5} 625\)</span> would be using this method. For this reason, we use exponents which are intimately related to logarithms.</p><p>A logarithm can be expressed identically using exponents for example:</p><p>$$ \log _{3} 9 = 2 \leftrightarrow 3^2 = 9</p><p>$$</p><p>By carrying out the conversion in stages, we can work out the answer to the question a logarithm poses.</p><p>Let’s work out <span class="math inline">\(\log \_{2} 8\)</span> using this method.</p><ol><li><p>First we add a variable (x) to the expression on the right hand:</p><p>$$ \log _{2} 8 \leftrightarrow x</p><p>$$</p></li><li><p>Next we take the base of the logarithm and combine it with x as an exponent. Now our formula looks like this:</p><p>$$ \log _{2} 8 \leftrightarrow 2^x</p><p>$$</p></li><li><p>Next we add an equals and the number that is left from the logarithm (8):</p></li></ol><p>$$ \log _{2} 8 \leftrightarrow 2^x = 8</p><p>$$</p><p>Then the problem is reduced to: how many times do you need to multiply two by itself to get 8? The answer is 3 : 2 x 2 x 2 or 2 p3. Hence we have the balanced equation:</p><p>$$ \log _{2} 8 \leftrightarrow 2^3 = 8</p><p>$$</p><h2 id="common-base-values">Common base values</h2><p>Often times a base won’t be specified in a log expression. For example:</p><p>$$ \log1000</p><p>$$</p><p>This is just a shorthand and it means that the base value is ten, i.e that the logarithm is written in denary (base 10). So the above actually means:</p><p>$$ \log _{10} 1000 = 3</p><p>$$</p><p>This is referred to as the <strong>common logarithm</strong></p><p>Another frequent base is Euler’s number (approx. 2.71828) known as the <strong>natural logarithm</strong></p><p>An example</p><p>$$ \log _{e} 7.389 = 2</p><p>$$</p></div></article><nav class="ui attached segment deemphasized backlinksPane" id="neuron-backlinks-pane"><h3 class="ui header">Backlinks</h3><ul class="backlinks"><li><span class="zettel-link-container cf"><span class="zettel-link"><a href="The_History_of_Computing_Swade.html">History of Computing (Swade, 2022 )</a></span></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc">Modern aids to calculation: slide rules following the discovery of <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Logarithms"><a href="Logarithms.html">logarithms</a></span></span></div></li></ul></li><li><span class="zettel-link-container cf"><span class="zettel-link"><a href="Memory_addresses.html">Memory addresses</a></span></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc"><p>We need to reverse this formula to find out how many bits we need to represent a given number of addresses. We can do this with a <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Logarithms"><a href="Logarithms.html">logarithm</a></span></span>.</p></div></li></ul></li></ul></nav><nav class="ui attached segment deemphasized bottomPane" id="neuron-tags-pane"><div><span class="ui basic label zettel-tag" title="Tag">algebra</span></div></nav><nav class="ui bottom attached icon compact inverted menu blue" id="neuron-nav-bar"><!--replace-start-9--><!--replace-end-9--><a class="right item" href="impulse.html" title="Open Impulse"><i class="wave square icon"></i></a></nav></div></div><!--replace-end-6--><!--replace-end-3--><!--replace-end-2--><div class="ui center aligned container footer-version"><div class="ui tiny image"><a href="https://neuron.zettel.page"><img alt="logo" src="https://raw.githubusercontent.com/srid/neuron/master/assets/neuron.svg" title="Generated by Neuron 1.9.35.3" /></a></div></div></div></body></html> |