https://peda.net/id/0cfa6da171e 2025-08-05T13:50:46+03:00 https://peda.net/:static/539/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/0cfac7ca71e 2018-03-07T09:02:04+02:00 <p><span><b>Määritelmä</b><br/> </span></p> <ul> <li>Olkoon<img src="https://math-demo.abitti.fi/math.svg?latex=a%3E0%7B%2C%7D%5C%20a%5Cne1%5C%20%5Ctext%7Bja%7D%5C%20b%3E0" alt="a&gt;0{,}\ a\ne1\ \text{ja}\ b&gt;0"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff; color: #333333; font-family: 'Times New Roman'; font-size: 17px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;"--><span>. </span><span>Logaritmi </span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ab" alt="\log_ab"/><span> tarkoittaa lukua<span> </span></span><em>x</em><span>, joka toteuttaa eksponenttiyhtälön </span><img src="https://math-demo.abitti.fi/math.svg?latex=a%5Ex%3Db" alt="a^x=b"/> eli</li> <li><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ab%3Dx%5C%20%5C%20%5C%20%5C%20%5CLeftrightarrow%5C%20%5C%20%5C%20%5C%20a%5Ex%3Db" alt="\log_ab=x\ \ \ \ \Leftrightarrow\ \ \ \ a^x=b"/></li> </ul> <div><hr class="float break"></hr></div> <div><b>Esim</b>. Ratkaise <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%3D8" alt="2^x=8"/> .<br/> <span class="editor underline">Tapa 0.</span> Toimii vain sopivilla luvuilla.<br/> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%3D8" alt="2^x=8"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%3D2%5E3" alt="2^x=2^3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D3" alt="x=3"/></div> <br/> <span class="editor underline">Tapa 1.</span> <br/> &quot;2-kantainen potenssi on 8, joten kysytty eksponentti <em>x</em> on 2-kantainen logaritmi 8:sta&quot;<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Clog_28" alt="x=\log_28"/><br/> Yhtälön ratkaisu on <em>x</em> = 3.<br/> <br/> <span class="editor underline">Tapa 2.</span> <br/> Potenssiinkorotus ja logatrimi &quot;kumoavat toisensa&quot;.<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%3D8" alt="2^x=8"/>  || <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_2%5Cleft(%5Cright)" alt="\log_2\left(\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_2%5Cleft(2%5Ex%5Cright)%3D%5Clog_2%5Cleft(8%5Cright)" alt="\log_2\left(2^x\right)=\log_2\left(8\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D3" alt="x=3"/> <div><br/> ja:</div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_2x%3D3" alt="\log_2x=3"/>  || <img src="https://math-demo.abitti.fi/math.svg?latex=2%5E%7B%5Cleft(%5Cright)%7D" alt="2^{\left(\right)}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5E%7B%5Clog_2x%7D%3D2%5E3" alt="2^{\log_2x}=2^3"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D8" alt="x=8"/></div> <br/> <hr class="float break"></hr><br/> <b>Esim 1</b>. Ratkaise<br/> a) <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%3D11" alt="2^x=11"/><br/> b) <img src="https://math-demo.abitti.fi/math.svg?latex=5%5Ccdot4%5E%7B3x%7D%3D30" alt="5\cdot4^{3x}=30"/> <br/> c) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_5x%3D4" alt="\log_5x=4"/><br/> <br/> <b>Esim 2</b>. Määritä<br/> a) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_327" alt="\log_327"/><br/> b) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_2%5Cfrac%7B1%7D%7B16%7D" alt="\log_2\frac{1}{16}"/><br/> c) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_41" alt="\log_41"/><br/> d) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_5%5Csqrt%7B5%7D" alt="\log_5\sqrt{5}"/><br/> <br/> <hr class="float break"></hr><br/> <b>Logaritmien laskusäännöt</b><br/> <div> <ul> <li><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_a%5Cleft(xy%5Cright)%3D%5Clog_ax%2B%5Clog_ay" alt="\log_a\left(xy\right)=\log_ax+\log_ay"/></li> <li><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_a%5Cleft(%5Cfrac%7Bx%7D%7By%7D%5Cright)%3D%5Clog_ax-%5Clog_ay" alt="\log_a\left(\frac{x}{y}\right)=\log_ax-\log_ay"/></li> <li><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ax%5Er%3Dr%5Clog_ax" alt="\log_ax^r=r\log_ax"/></li> </ul> <hr class="float break"></hr><br/> <b>Esim 3. </b><br/> Sievennä<br/> a) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_672-%5Clog_62" alt="\log_672-\log_62"/><br/> b) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_2%5Csqrt%5B3%5D%7B16%7D" alt="\log_2\sqrt[3]{16}"/><br/> <br/> </div> </div> 2025-08-05T13:50:46+03:00 https://peda.net/id/0cfb2f8b71e 2018-03-07T09:13:19+02:00 <span>a)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%3D11" alt="2^x=11"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Clog_211%5C%20%5Cleft(%5Capprox3.459...%5Cright)" alt="x=\log_211\ \left(\approx3.459...\right)"/> <div>b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5%5Ccdot4%5E%7B3x%7D%3D30" alt="5\cdot4^{3x}=30"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=4%5E%7B3x%7D%3D6" alt="4^{3x}=6"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D%5Clog_46" alt="3x=\log_46"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Clog_46%7D%7B3%7D%5C%20%5C%20%5Cleft(%5Capprox0%7B%2C%7D4308...%5Cright)" alt="x=\frac{\log_46}{3}\ \ \left(\approx0{,}4308...\right)"/> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_5x%3D4" alt="\log_5x=4"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D5%5E4" alt="x=5^4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D625" alt="x=625"/> <div> </div> <div>2. a) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_327%3D3" alt="\log_327=3"/> || <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Ex%3D27" alt="3^x=27"/></div> <div> <div>b) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_2%5Cfrac%7B1%7D%7B16%7D%3D-4" alt="\log_2\frac{1}{16}=-4"/>  || <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ex%3D%5Cfrac%7B1%7D%7B16%7D%3D%5Cfrac%7B1%7D%7B2%5E4%7D%3D2%5E%7B-4%7D" alt="2^x=\frac{1}{16}=\frac{1}{2^4}=2^{-4}"/></div> c) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_41%3D0" alt="\log_41=0"/> || 4^0 = 1</div> <div>d) </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_5%5Csqrt%7B5%7D%3D%5Clog_55%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="\log_5\sqrt{5}=\log_55^{\frac{1}{2}}=\frac{1}{2}"/><br/> <div> </div> <div>esim 3.</div> <div>a) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_672-%5Clog_62%3D%5Clog_6%5Cleft(%5Cfrac%7B72%7D%7B2%7D%5Cright)%3D%5Clog_636%3D2" alt="\log_672-\log_62=\log_6\left(\frac{72}{2}\right)=\log_636=2"/></div> <div>b) </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_2%5Csqrt%5B3%5D%7B16%7D%3D%5Clog_216%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%3D%5Clog_2%5Cleft(2%5E4%5Cright)%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%3D%5Clog_2%5Cleft(2%5E%7B%5Cfrac%7B4%7D%7B3%7D%7D%5Cright)%3D%5Cfrac%7B4%7D%7B3%7D" alt="\log_2\sqrt[3]{16}=\log_216^{\frac{1}{3}}=\log_2\left(2^4\right)^{\frac{1}{3}}=\log_2\left(2^{\frac{4}{3}}\right)=\frac{4}{3}"/><br/> <br/> <div> </div> 2025-08-05T13:50:46+03:00