https://peda.net/id/0cfb918471e 2025-08-05T13:50:46+03:00 https://peda.net/:static/533/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/0cfbee4171e 2018-03-14T09:26:00+02:00 <div><b>423</b><br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3De%5E%7B2x%7D-25" alt="f\left(x\right)=e^{2x}-25"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D0" alt="f\left(x\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B2x%7D-25%3D0" alt="e^{2x}-25=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B2x%7D%3D25" alt="e^{2x}=25"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(e%5Ex%5Cright)%5E2%3D25" alt="\left(e^x\right)^2=25"/><br/> kolme tapaa jatkaa:<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Barray%7D%7Bl%7Cl%7D%0Ae%5Ex%3D5%5C%20%5C%20%5C%20%5C%20%5Ctext%7Btai%7D%5C%20%5C%20%5C%20e%5Ex%3D-5%26%5Cln25%3D2x%26%5Cln5%5E2%3D2x%5C%5C%0A%5Chline%0Ax%3D%5Cln5%5C%20%5C%20%5Ctext%7Btai%7D%5C%20ei%5C%20ratk.%26x%3D%5Cfrac%7B%5Cln25%7D%7B2%7D%3D%5Cfrac%7B%5Cln5%5E2%7D%7B2%7D%3D%5Cfrac%7B2%5Cln5%7D%7B2%7D%3D%5Cln5%262%5Cln5%3D2x%5C%5C%0A%26%26%5Cln5%3Dx%0A%5Cend%7Barray%7D" alt="\begin{array}{l|l}&#10;e^x=5\ \ \ \ \text{tai}\ \ \ e^x=-5&amp;\ln25=2x&amp;\ln5^2=2x\\&#10;\hline&#10;x=\ln5\ \ \text{tai}\ ei\ ratk.&amp;x=\frac{\ln25}{2}=\frac{\ln5^2}{2}=\frac{2\ln5}{2}=\ln5&amp;2\ln5=2x\\&#10;&amp;&amp;\ln5=x&#10;\end{array}"/><br/> <br/> 2025-08-05T13:50:46+03:00 https://peda.net/id/0cfc526071e 2018-03-09T14:09:06+02:00 <ul> <li><em>Määritelmä: <br/> </em>Luvun <em>b &gt; 0</em> <em>e</em>-kantaista logaritmia <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_eb" alt="\log_eb"/> kutsutaan luonnolliseksi logaritmiksi ja merkitään <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%20b" alt="\ln b"/><br/> eli <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_eb%3Dx%5C%20%5C%20%5C%20%5CLeftrightarrow%5C%20%5C%20%5C%20e%5Ex%3Db" alt="\log_eb=x\ \ \ \Leftrightarrow\ \ \ e^x=b"/> voidaan kirjoittaa<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%20b%3Dx%5C%20%5C%20%5C%20%5CLeftrightarrow%5C%20%5C%20%5C%20e%5Ex%3Db" alt="\ln b=x\ \ \ \Leftrightarrow\ \ \ e^x=b"/></li> </ul> <b><span class="editor underline">ESIM 1</span></b><br/> a) <span><span> 3</span></span><em>e</em>^<em>x</em><span><span> </span>– 9 = 0<br/> </span>b) <span>ln<span> </span></span><em>x</em><span><span> </span>= –2<br/> c) </span><em> e</em>^<em>3x</em><span><span> </span>= 8<br/> </span>d) <em>e</em>^2<em>x</em><span><span> </span>– 2</span><em>e</em>^<em>x</em><span><span> </span>= 0<br/> </span> <ul> <li><em>Lause</em>: Positiivisille luvuille <em>a</em> ja <em>b</em> pätee seuraavat säännöt:<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ea%3De%5Eb%5C%20%5C%20%5CLeftrightarrow%5C%20%5C%20%5C%20a%5C%20%3Db" alt="e^a=e^b\ \ \Leftrightarrow\ \ \ a\ =b"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%20a%3D%5Cln%20b%5C%20%5C%20%5CLeftrightarrow%5C%20%5C%20a%5C%20%3D%5C%20b" alt="\ln a=\ln b\ \ \Leftrightarrow\ \ a\ =\ b"/></li> </ul> <p><br/> <b><span class="editor underline">ESIM 2.</span></b><br/> Ratkaise <span>0,63</span>^<em>x</em><span> ⋅ 4,2 = 2. Anna vastauksena kaksidesimaalinen likiarvo.<br/> <br/> </span></p> <p><span><span class="editor underline"><b>Eksponenttifunktion derivaatta</b></span><br/> </span></p> <ul> <li><span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7BD%7Da%5Ex%3Da%5Ex%5Cln%20a" alt="\text{D}a^x=a^x\ln a"/><span> ,</span><img src="https://math-demo.abitti.fi/math.svg?latex=a%3E0" alt="a&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;"--><br/> </span></li> </ul> ESIM. <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7BD%7D5%5Ex%3D5%5Ex%5Cln5" alt="\text{D}5^x=5^x\ln5"/> 2025-08-05T13:50:46+03:00 https://peda.net/id/0cfcae4b71e 2018-03-09T14:08:51+02:00 <b>ESIM 1</b><br/> <br/> <span>a) </span><img src="https://math-demo.abitti.fi/math.svg?latex=3e%5Ex-9%3D0" alt="3e^x-9=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3e%5Ex%3D9" alt="3e^x=9"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%3D3" alt="e^x=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cln3" alt="x=\ln3"/><br/> <span>b)</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%20x%3D-2" alt="\ln x=-2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B-2%7D%3Dx" alt="e^{-2}=x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%7D%7Be%5Ex%7D" alt="x=\frac{1}{e^x}"/> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B3x%7D%3D27" alt="e^{3x}=27"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D%5Cln8" alt="3x=\ln8"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D%5Cln2%5E3" alt="3x=\ln2^3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D3%5Cln2" alt="3x=3\ln2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cln2" alt="x=\ln2"/><br/> <br/> d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5E%7B2x%7D-2e%5Ex%3D0" alt="e^{2x}-2e^x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5Exe%5Ex-2e%5Ex%3D0" alt="e^xe^x-2e^x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%5Cleft(e%5Ex-2%5Cright)%3D0" alt="e^x\left(e^x-2\right)=0"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%3D0%5C%20%5C%20%5C%20%5Ctext%7Btai%7D%5C%20%5C%20e%5Ex-2%3D0" alt="e^x=0\ \ \ \text{tai}\ \ e^x-2=0"/></div> <div>ei ratkaisua tai <span> </span><img src="https://math-demo.abitti.fi/math.svg?latex=e%5Ex%3D2" alt="e^x=2"/></div>  <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cln2" alt="x=\ln2"/><br/> <br/> <b>ESIM2</b><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D63%5Ex%5Ccdot4%7B%2C%7D2%3D2" alt="0{,}63^x\cdot4{,}2=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D63%5Ex%3D%5Cfrac%7B2%7D%7B4%7B%2C%7D2%7D" alt="0{,}63^x=\frac{2}{4{,}2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5Cleft(0%7B%2C%7D63%5Ex%5Cright)%3D%5Cln%5Cleft(%5Cfrac%7B2%7D%7B4%7B%2C%7D2%7D%5Cright)" alt="\ln\left(0{,}63^x\right)=\ln\left(\frac{2}{4{,}2}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cln%5Cleft(0%7B%2C%7D63%5Cright)%3D%5Cln%5Cleft(%5Cfrac%7B2%7D%7B4%7B%2C%7D2%7D%5Cright)" alt="x\ln\left(0{,}63\right)=\ln\left(\frac{2}{4{,}2}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cln%5Cleft(%5Cfrac%7B2%7D%7B4%7B%2C%7D2%7D%5Cright)%7D%7B%5Cln%5Cleft(0%7B%2C%7D63%5Cright)%7D%3D1%7B%2C%7D6058...%5Capprox1%7B%2C%7D61" alt="x=\frac{\ln\left(\frac{2}{4{,}2}\right)}{\ln\left(0{,}63\right)}=1{,}6058...\approx1{,}61"/><br/> <br/> <br/> HUOM! Myös <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Clog_%7B0%7B%2C%7D63%7D%5Cleft(%5Cfrac%7B2%7D%7B4%7B%2C%7D2%7D%5Cright)" alt="x=\log_{0{,}63}\left(\frac{2}{4{,}2}\right)"/> on oikein eli <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_%7B0%7B%2C%7D63%7D%5Cleft(%5Cfrac%7B2%7D%7B4%7B%2C%7D2%7D%5Cright)%3D%5Cfrac%7B%5Cln%5Cleft(%5Cfrac%7B2%7D%7B4%7B%2C%7D2%7D%5Cright)%7D%7B%5Cln0%7B%2C%7D63%7D" alt="\log_{0{,}63}\left(\frac{2}{4{,}2}\right)=\frac{\ln\left(\frac{2}{4{,}2}\right)}{\ln0{,}63}"/>.<br/> <br/> <br/> Yleisestikin, kantalukua voi vaihtaa: <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_ab%3D%5Cfrac%7B%5Clog_cb%7D%7B%5Clog_ca%7D" alt="\log_ab=\frac{\log_cb}{\log_ca}"/> (totea)<br/> <br/> </div> </div> </div> 2025-08-05T13:50:46+03:00