https://peda.net/id/0cfd63d071e 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/0cfdc2d171e 2018-03-10T14:29:42+02:00 <ul> <li>Potenssifunktio <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Da%5Ex" alt="f\left(x\right)=a^x"/> ja logaritmifunktio <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Clog_ax" alt="f\left(x\right)=\log_ax"/> ovat toistensa ns. käänteisfunktioita (kuvaajat peilikuvia suoran y = x suhteen). <ul> <li>Esim. a^x:n pistettä (3, 5) vastaa log_a x:n piste (5,3)</li> </ul> </li> </ul> <ul> <li><em>Lause</em>: Logaritmifunktio <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Clog_ax" alt="f\left(x\right)=\log_ax"/> on jatkuva, kun a &gt; 0 ja <ul> <li>kasvava, kun a &gt; 1</li> <li>vähenevä, kun 0 &lt; a &lt; 1</li> </ul> </li> </ul> <br/> <b><span class="editor underline">Esim 1<br/> </span></b>Milloin <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Clog_3%5Cleft(x-1%5Cright)" alt="f\left(x\right)=\log_3\left(x-1\right)"/> on määritelty? Määritä nollakohdat.<br/> <br/> <b><span class="editor underline">Esim 2</span></b><br/> Ratkaise <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cln%20x%3D%5Cln%5Cleft(2x-1%5Cright)" alt="2\ln x=\ln\left(2x-1\right)"/>.<br/> <div><br/> <b><span class="editor underline">Esim 3</span></b><br/> <span>Määritä funktioiden </span><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Clg%5Cleft(x%2B4%5Cright)" alt="f\left(x\right)=\lg\left(x+4\right)"/><span>ja</span><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D2-%5Clg%5Cleft(x-4%5Cright)" alt="g\left(x\right)=2-\lg\left(x-4\right)"/><span> leikkauspisteet.</span><br/> <br/> </div> <div> </div> 2025-08-05T13:50:46+03:00 https://peda.net/id/0cfe29d471e 2018-03-13T11:54:42+02:00 <b><span class="editor underline">ESIM 1</span></b><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Clog_3%5Cleft(x-1%5Cright)" alt="f\left(x\right)=\log_3\left(x-1\right)"/><br/> Määritelty, kun x-1&gt;0 eli kun x&gt;1.<br/> <br/> Nollakohdat:<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_3%5Cleft(x-1%5Cright)%3D0" alt="\log_3\left(x-1\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5E%7B%5Clog_3%5Cleft(x-1%5Cright)%7D%3D3%5E0" alt="3^{\log_3\left(x-1\right)}=3^0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-1%3D1" alt="x-1=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2" alt="x=2"/><br/> <br/> <span class="editor underline"><b>ESIM 2</b></span><br/> <div>Vasen puoli määritelty, kun x &gt; 0, oikea puoli, kun 2x-1&gt;0 eli x&gt;½. Vastauksen pitää siis olla x &gt; ½.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cln%20x%3D%5Cln%5Cleft(2x-1%5Cright)" alt="2\ln x=\ln\left(2x-1\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%20x%5E2%3D%5Cln%5Cleft(2x-1%5Cright)" alt="\ln x^2=\ln\left(2x-1\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D2x-1" alt="x^2=2x-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-2x%2B1%3D0" alt="x^2-2x+1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B2%5Cpm%5Csqrt%7B4-4%5Ccdot1%5Ccdot1%7D%7D%7B2%7D" alt="x=\frac{2\pm\sqrt{4-4\cdot1\cdot1}}{2}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B2%5Cpm%5Csqrt%7B0%7D%7D%7B2%7D%3D1" alt="x=\frac{2\pm\sqrt{0}}{2}=1"/></div> <br/> <br/> <b><span class="editor underline">ESIM 3</span></b><br/> Määritä funktioiden <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Clg%5Cleft(x%2B4%5Cright)" alt="f\left(x\right)=\lg\left(x+4\right)"/>ja<img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D2-%5Clg%5Cleft(x-4%5Cright)" alt="g\left(x\right)=2-\lg\left(x-4\right)"/> leikkauspisteet. <div> </div> <div>Funktio f(x) määritelty, kun x &gt; -4 ja g(x), kun x &gt; 4. Ratkaisu voi siis löytyä, jos x &gt; 4.</div> <div><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dg%5Cleft(x%5Cright)" alt="f\left(x\right)=g\left(x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clg%5Cleft(x%2B4%5Cright)%3D2-%5Clg%5Cleft(x-4%5Cright)" alt="\lg\left(x+4\right)=2-\lg\left(x-4\right)"/> || pitäisi saada lg(jtn1) = lg(jtn2)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clg%5Cleft(x%2B4%5Cright)%2B%5Clg%5Cleft(x-4%5Cright)%3D2" alt="\lg\left(x+4\right)+\lg\left(x-4\right)=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clg%5Cleft(%5Cleft(x%2B4%5Cright)%5Cleft(x-4%5Cright)%5Cright)%3D%5Clg100" alt="\lg\left(\left(x+4\right)\left(x-4\right)\right)=\lg100"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2-16%3D100" alt="x^2-16=100"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D116" alt="x^2=116"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpm%5Csqrt%7B116%7D" alt="x=\pm\sqrt{116}"/><span>  || Negatiivinen ei käy, x&gt;4</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Csqrt%7B116%7D%3D2%5Csqrt%7B29%7D" alt="x=\sqrt{116}=2\sqrt{29}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y%3Df%5Cleft(2%5Csqrt%7B29%7D%5Cright)%3D%5Clg%5Cleft(2%5Csqrt%7B29%7D%2B4%5Cright)" alt="y=f\left(2\sqrt{29}\right)=\lg\left(2\sqrt{29}+4\right)"/></div> <div> </div> <div><span>V: Leikkauspiste on </span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5C%20%5Csqrt%7B116%7D%7B%2C%7D%5C%20%5C%20%5C%20%5C%20%5Clg%5Cleft(%5Csqrt%7B116%7D%2B4%5Cright)%5Cright)" alt="\left(\ \sqrt{116}{,}\ \ \ \ \lg\left(\sqrt{116}+4\right)\right)"/><span>.</span></div> </div> </div> 2025-08-05T13:50:46+03:00 https://peda.net/id/0cff322171e 2018-03-14T09:12:03+02:00 456.<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cln%5Cleft(5x-1%5Cright)%3D%5Cln%5Cleft(2x%5Cright)" alt="\ln\left(5x-1\right)=\ln\left(2x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7BM%C3%A4%C3%A4rittelyjoukko%3A%20%7D%5C%205x-1%3E0%5C%20%5Ctext%7Bja%7D%5C%202x%3E0" alt="\text{Määrittelyjoukko: }\ 5x-1&gt;0\ \text{ja}\ 2x&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/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E%5Cfrac%7B1%7D%7B5%7D" alt="x&gt;\frac{1}{5}"/><!--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;"--> <div><img src="https://math-demo.abitti.fi/math.svg?latex=5x-1%3D2x" alt="5x-1=2x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D1" alt="3x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B1%7D%7B3%7D" alt="x=\frac{1}{3}"/><br/> <div> </div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Clog_2x%3D%5Clog_2%5Cleft(3x%2B4%5Cright)" alt="2\log_2x=\log_2\left(3x+4\right)"/></div> <span>Määrittelyjoukko:</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E0%5C%20%5Ctext%7Bja%7D%5C%203x%2B4%3E0%5C%20%5Ctext%7Beli%7D%5C%20x%3E-%5Cfrac%7B4%7D%7B3%7D" alt="x&gt;0\ \text{ja}\ 3x+4&gt;0\ \text{eli}\ x&gt;-\frac{4}{3}"/><!--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/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3E0" alt="x&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;"--> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Clog_2x%3D%5Clog_2%5Cleft(3x%2B4%5Cright)" alt="2\log_2x=\log_2\left(3x+4\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clog_2x%5E2%3D%5Clog_2%5Cleft(3x%2B4%5Cright)" alt="\log_2x^2=\log_2\left(3x+4\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D3x%2B4" alt="x^2=3x+4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(x%3D-1%5Cright)%5C%20%5Ctext%7Btai%7D%5C%20x%3D4" alt="\left(x=-1\right)\ \text{tai}\ x=4"/> <div><span class="editor underline">v: x = 4</span></div> <div> </div> <div> </div> <div> </div> <div>-------------------------</div> <div>D ln x = 1/x todistus:<br/> <br/> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cln%20x" alt="f\left(x\right)=\ln x"/></div> <div>Derivoidaan kohdassa x = x_0</div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x_0%5Cright)%3D%5Clim_%7Bx%5Crightarrow%7Dx_0%5Cleft(%5Cfrac%7B%5Cln%20x-%5Cln%20x_0%7D%7Bx-x_0%7D%5Cright)" alt="f'\left(x_0\right)=\lim_{x\rightarrow}x_0\left(\frac{\ln x-\ln x_0}{x-x_0}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Cln%20x%7B%2C%7D%5C%20%5C%20%5C%20y_0%3D%5Cln%20x_0%7B%2C%7D%5C%20%5C%20%5Ctext%7Bjoten%7D%5C%20%5C%20x%3De%5Ey%5C%20%5Ctext%7Bja%7D%5C%20x_0%3De%5E%7By_0%7D" alt="y=\ln x{,}\ \ \ y_0=\ln x_0{,}\ \ \text{joten}\ \ x=e^y\ \text{ja}\ x_0=e^{y_0}"/><br/> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D%5Clim_%7Bx%5Crightarrow%20x_0%7D%5Cleft(%5Cfrac%7B%5Cln%20x-%5Cln%20x_0%7D%7Bx-x_0%7D%5Cright)%3D%5Clim_%7Bx%5Crightarrow%20x_0%7D%5Cleft(%5Cfrac%7By-y_0%7D%7Be%5Ey-e%5E%7By_0%7D%7D%5Cright)%3D%5Clim_%7Bx%5Crightarrow%20x_0%7D%5Cleft(%5Cfrac%7B1%7D%7B%5Cfrac%7Be%5Ey-e%5E%7By_0%7D%7D%7By-y_0%7D%7D%5Cright)" alt="f'\left(x\right)=\lim_{x\rightarrow x_0}\left(\frac{\ln x-\ln x_0}{x-x_0}\right)=\lim_{x\rightarrow x_0}\left(\frac{y-y_0}{e^y-e^{y_0}}\right)=\lim_{x\rightarrow x_0}\left(\frac{1}{\frac{e^y-e^{y_0}}{y-y_0}}\right)"/>  || <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clim_%7By%5Crightarrow%20y_0%7D%5Cleft(%5Cfrac%7Be%5Ey-e%5E%7By_0%7D%7D%7By-y_0%7D%5Cright)%3De%5E%7By_0%7D" alt="\lim_{y\rightarrow y_0}\left(\frac{e^y-e^{y_0}}{y-y_0}\right)=e^{y_0}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Clim_%7Bx%5Cto%20x_0%7D%5Cleft(%5Cfrac%7B1%7D%7Be%5E%7By_0%7D%7D%5Cright)%3D%5Clim_%7Bx%5Cto%20x_0%7D%5Cleft(%5Cfrac%7B1%7D%7Bx_0%7D%5Cright)%3D%5Cfrac%7B1%7D%7Bx_0%7D" alt="=\lim_{x\to x_0}\left(\frac{1}{e^{y_0}}\right)=\lim_{x\to x_0}\left(\frac{1}{x_0}\right)=\frac{1}{x_0}"/></div> 2025-08-05T13:50:46+03:00