(a) \(\displaystyle f(x)=3-3^x\)
(b) \(\displaystyle f(x)=3^{3-x}\)
(c) \(\displaystyle f(x)=\sqrt{3}^{\,x+2}-1\)
(d) \(\displaystyle f(x)=e^{-x}-3\)
Opomba: Število \(e\doteq2,\!71828\) se imenuje Eulerjevo število. Funkcija \(f(x)=e^x\) se imenuje naravna eksponentna funkcija.(a) Izračunaj osnovo \(a\) in nariši graf te funkcije.
(b) Za dobljeno vrednost \(a\) nariši še graf funkcije \(y=-\,a^{2x+6}\).
Rešitev: (a) \(a=\sqrt{2}\)(a) \(3^{x+2}-2\cdot 3^{x+1}-3^x\)
(b) \({\displaystyle\frac{7^{x+1}-7^{x-1}}{7^{x+1}+7^x}}\)
(c) \({\displaystyle\frac{12^{x+1}-3\cdot 12^x}{2^{2x+2}-2^{2x+1}-2^{2x}}}\)
Rešitev: (a) \(\cdots=2\cdot 3^x\), (b) \(\cdots=\frac{6}{7}\), (c) \(\cdots=3^{x+2}\) (Namig: Pomagaj si z izpostavljanjem skupnega faktorja.)(a) \(\displaystyle\frac{2^{x+5}+2^{x+1}+7\cdot2^x}{2^{x+2}+2^x+2^{x-3}}\)
(b) \(\displaystyle\frac{2^{3x+1}+2^{3x-1}-5\cdot2^{3x-3}}{4^{x+1}-4^{x-1}}\)
(c) \(\displaystyle\frac{6^x+3\cdot6^{x-1}-2\cdot6^{x-2}}{3^{x-1}+3^{x-2}+3^{x-3}}\)
(d) \(\displaystyle\frac{a^{x+2}-a^{x+1}-2a^x}{a^{x+2}-2a^{x+1}}\)
Rešitev: (a) \(\cdots=8\), (b) \(\cdots=2^{x-1}\), (c) \(\cdots=2^x\cdot3\), (d) \(\cdots=\frac{a+1}{a}\)(a) \(\displaystyle\frac{e^{2x}+1}{e^x+\frac{1}{e^x}}\)
(b) \(\displaystyle\frac{e^x+e^{-x}}{e^{1+x}+e^{1-x}}\)
Rešitev: (a) \(\cdots=e^x\), (b) \(\cdots=\frac{1}{e}\)(a) \({\displaystyle 5^{3x-4}=25}\)
(b) \({\displaystyle 2^{x-1}=\frac{4}{\sqrt{2}}}\)
(c) \(3^{x+2}\cdot \sqrt{3}^{~x}\cdot \Big(\frac{\textstyle 1}{\textstyle 9}\Big)^{x-2}=1\)
Rešitev: (a) \(x=2\), (b) \(x=\frac{5}{2}\), (c) \(x=12\)(a) \(\displaystyle 2^{3x-6}=5^{3x-6} \)
(b) \(\displaystyle 9^x\cdot\frac{1}{3}=4^{2x}\cdot\frac{1}{4}\)
(c) \({\displaystyle 3^x=5^{x-1}\cdot \frac{27}{25}}\)
Rešitev: (a) \(x=2\), (b) \(x=\frac{1}{2}\), (c) \(x=3\)(a) \({\displaystyle 7^{x+2}+7^x=2450}\)
(b) \({\displaystyle \frac{2^{x+1}+2^x}{16-2^{x-1}}=6}\)
(c) \({\displaystyle \frac{3^{x+1}-4^x+3^{x-1}}{5\cdot 4^{x-1} - 2\cdot 3^x}=1}\)
Rešitev: (a) \(x=2\), (b) \(x=4\), (c) \(x=3\)(a) \(\displaystyle 9^{2x}-2\cdot 9^x=3\)
(b) \(\displaystyle 3^x-6=\frac{27}{3^x}\)
(c) \(\displaystyle 2^x-2^{5-x}=4\)
Rešitev: (a) \(x=\frac{1}{2}\), (b) \(x=2\), (c) \(x=3\) (Namig: Pri vseh treh enačbah si lahko pomagaš z uvedbo nove neznanke.)(a) \(\displaystyle 4^{x+1}=2\sqrt{2}\)
(b) \(\displaystyle 8\cdot 3^{x-1}=60+4\cdot3^{x-2}\)
(c) \(\displaystyle \frac{4^{x-1}}{4}+2^x\cdot 2^{x-4}=16\)
(d) \(\sqrt{3}^{\,x+1}\cdot 27^{\,x-3}\cdot \left(\frac{\textstyle 1}{\textstyle 9}\right)^{x}=\frac{\textstyle 1}{\textstyle 3}\)
Rešitev: (a) \(x=-\frac{1}{4}\), (b) \(x=3\), (c) \(x=\frac{7}{2}\), (d) \(x=5\)(a) \(\displaystyle 3^{2x+1}=3^{x+2}+162\)
(b) \(5\cdot3\raise0.8em{\frac{x}{2}}+2\raise0.8em{\frac{x}{2}}=\sqrt{3}^{\,x+4}-\sqrt{2}^{\,x+6}\)
(c) \(\displaystyle 5^{2x-13}-5^{2-3x}=0\)
(d) \(\displaystyle 2^{2x+3}=5^{x+2}-3\cdot 4^{x+1}\)
Rešitev: (a) \(x=2\), (b) \(x=4\), (c) \(x=3\), (d) \(x=-1\)(a) \(\displaystyle 9^x+3\cdot4^x=9^{x+1}-4^{x+2}-2^{2x+3}\)
(b) \(\displaystyle 5^{x+2}-2^{3x+5}=5^{x}+8^{x}-8^{x+1}-8\cdot5^{x+1}\)
(c) \(\displaystyle 9^{x+1}+26\cdot3^x=3\)
(d) \(\displaystyle \frac{\big(2^x\big)^{x+3}\cdot0,\!25^{x+1}}{\sqrt{2}^{~x+2}}=1\)
Rešitev: (a) \(x=\frac{3}{2}\), (b) \(x=2\), (c) \(x=-2\), (d) \(x_1=\frac{3}{2},~ x_2=-2\)(a) \(\log_3 81\)
(b) \(\log_4 32\)
(c) \(\log_5 \frac{1}{25}\)
(d) \(\log_{10}1\,000\,000\)
Rešitev: (a) \(\cdots=4\), (b) \(\cdots=\frac{5}{2}\), (c) \(\cdots=-2\), (d) \(\cdots=6\)(a) \(\log_{16} \sqrt{2}\)
(b) \(\log_9 27\sqrt{3}\)
(c) \(\log\!\raise-0.4em{\scriptstyle\sqrt{5}}~ \Big(\frac{\textstyle 1}{\textstyle 5\sqrt{5}}\Big)\)
(d) \(\log\!\raise-0.8em{\textstyle\frac{1}{8}}~ \sqrt[\scriptstyle 3]{2}\)
Rešitev: (a) \(\cdots=\frac{1}{8}\), (b) \(\cdots=\frac{7}{4}\), (c) \(\cdots=-3\), (d) \(\cdots=-\frac{1}{9}\)(a) \(\displaystyle 5^{x-1}=\frac{\textstyle 7}{\textstyle 33}\)
(b) \(\displaystyle 3^{2x+1}=1234\)
(c) \(\Big(\frac{\textstyle 2}{\textstyle 3}\Big)^{3x}=\frac{\textstyle 125}{\textstyle 9}\)
(d) \(\displaystyle e^{x^2-1}=100\)
Rešitev: (a) \(x\doteq0,\!03656\), (b) \(x\doteq2,\!740\), (c) \(x\doteq-2,\!163\), (d) \(x_1\doteq-2,\!368,~ x_2\doteq2,\!368\)(a) \(\displaystyle 3^{x+1}+3^{x-1}=333\)
(b) \(\displaystyle 5^x=13\cdot2^x\)
(c) \(\displaystyle 7\cdot 4^{x-2}=5\cdot 3^{x+1}\)
(d) \(\displaystyle 2^{x}+3^{x}=2^{x+2}-3^{x+3}\)
Rešitev: (a) \(x\doteq4,\!1909\), (b) \(x\doteq2,\!7993\), (c) \(x\doteq12,\!2869\), (d) \(x\doteq-5,\!509\)(a) \(f(x)=\log_2(x+4)-2\)
(b) \(f(x)=2-\log_3(3x)\)
(c) \(f(x)=\ln(x+2)\)
(d) \(f(x)=\log\!\raise-0.8em{\textstyle\frac{1}{2}} (2x+6)\)
Opomba: Logaritem, ki ima za osnovo Eulerjevo število \(e\doteq2,\!71828\), imenujemo naravni logaritem in ga označimo kot \(\ln\). Torej: \(\ln x=\log_e x\).(a) določi definicijsko območje funkcije,
(b) ugotovi, za katere \(x\) velja: \(f(x)\leqslant 0\).
Rešitev: (a) \({\cal D}_f=(-1,\infty)\), (b) to velja za \(x\in (-1,1]\).(a) Določi osnovo \(a\).
(b) Nariši graf funkcije \(f\).
(c) Nariši še graf funkcije \(g(x)=f(6-2x)\).
Rešitev: (a) \(a=\frac{1}{4}\)(a) \(\log_2 (a^2 b \sqrt{2})\)
(b) \({\displaystyle\log \sqrt[\scriptstyle 3]{\frac{a^6 b^{12}}{c}}}\)
(c) \({\displaystyle\log\frac{100a}{\sqrt{bc^3}}}\)
Rešitev: (a) \(\cdots=\frac{1}{2}+2\log_2 a+\log_2 b\), (b) \(\cdots=2\log a+4\log b -\frac{1}{3}\log c\), (c) \(\cdots=2+\log a - \frac{1}{2}\log b-\frac{3}{2}\log c\)(a) \(\displaystyle \log \frac{\sqrt{a}}{bc}\)
(b) \(\log_a bc\)
Rešitev: (a) \(\cdots=\frac{5}{3}\), (b) \(\cdots=\frac{1}{6}\)(a) \(\log_a \frac{\textstyle 1}{\textstyle b}\)
(b) \(\log\!\lower0.8em{\textstyle\frac{1}{a}} \,\frac{\textstyle 1}{\textstyle b}\)
(c) \(\log_a ab\)
(d) \(\log_b a\)
Rešitev: (a) \(\cdots=-\frac{2}{3}\), (b) \(\cdots=\frac{2}{3}\), (c) \(\cdots=\frac{5}{3}\), (d) \(\cdots=\frac{3}{2}\)(a) \({\displaystyle \log a-\frac{2\log b+3\log c}{6}}\)
(b) \(3-\log_3 a+\frac{1}{3}\log_3 b-2\log_3 c\)
(c) \(\log(6a-6)-\log 2 -(\log(a-1)-\log a)\)
Rešitev: (a) \(\cdots=\log\frac{a}{\sqrt[6]{b^2c^3}}\), (b) \(\cdots=\log_3 \frac{27\sqrt[3]{b}}{ac^2}\), (c) \(\cdots=\log 3a\)(a) \(\log_3 x=\frac{1}{2}+\log_3 a+\log_3 b-\log_3 (a+b)\)
(b) \(\log_2 x= 3+\log_2 (a^2b-ab^2)- \log_2 (a^2-b^2)- \log_2 ab\)
(c) \(\ln x=5\ln 2+\ln(a^2-b^2)-3\ln 4-\ln (a-b)\)
Rešitev: (a) \(x=\frac{ab\sqrt{3}}{a+b}\), (b) \(x=\frac{8}{a+b}\), (c) \(x=\frac{a+b}{2}\)(a) \(\log x=2\log(a-b)-\log(a^2-b^2)\)
(b) \(\displaystyle \log y=2+2\log\frac{a}{2}-\frac{\log b+\log c}{2}\)
(c) \(\displaystyle \log_4 z=\frac{\log_2 a-3\log_2 b+5\log_2 c}{4}\)
Rešitev: (a) \(x=\frac{a-b}{a+b}\), (b) \(y=\frac{25a^2}{\sqrt{bc}}\), (c) \(z=\sqrt{\frac{ac^5}{b^3}}\)(a) \(\log_9 (x-3)=\frac{3}{2}\)
(b) \(\log_x 64=3\)
(c) \(\log_x (2x+15)=2\)
Rešitev: (a) \(x=30\), (b) \(x=4\), (c) \(x=5\) (Namig: Lahko si pomagaš z definicijo logaritma: \(\log_a b=c~\Longleftrightarrow~ a^c=b\))(a) \(\log_3(x+1)+\log_3(x-2)=\log_3(3x+10)\)
(b) \(1+\log_2(x-1)=2\log_2(x+3)-\log_2 x\)
(c) \(\log_8 \big(x-\frac{1}{3}\big)=\log_8 x-\frac{1}{3}\)
(d) \(\log_3 \sqrt{23x+12}=\log_3 x+1\)
Rešitev: (a) \(x=6\), (b) \(x=9\), (c) \(x=\frac{2}{3}\), (d) \(x=3\)(a) \({\displaystyle \log\Big(\frac{x-7}{2}\Big)=\frac{\log(x-7)}{2}}\)
(b) \({\displaystyle\frac{\log x+2}{\log(2x+12)}=2}\)
(c) \({\displaystyle\frac{\log (x-5)}{\log(7-x)}=\frac{1}{\,2\,}}\)
(d) \({\displaystyle\frac{\log(7x-3)+\log(x-1)}{\log 2x}=2}\)
Rešitev: (a) \(x=11\), (b) \(x_1=9,~ x_2=4\), (c) enačba nima rešitve, (d) \(x=3\)(a) \({\displaystyle 2\log x+1=\frac{1}{\log x}}\)
(b) \({\displaystyle\frac{\log x}{\log x+3}=\frac{4}{\log x}}\)
(c) \({\displaystyle\frac{\log x}{\log 10x}=\frac{\log 100x}{\log 10000x}}\)
Rešitev: (a) \(x_1=\frac{1}{10},~ x_2=\sqrt{10}\), (b) \(x_1=\frac{1}{100},~ x_2=1\,000\,000\), (c) \(x=100\) (Namig: Lahko si pomagaš z uvedbo nove neznanke.)(a) \(\log_4(x-3)=\log_{16}(2x+2)\)
(b) \(\log_2(x-2)=\log_8(7x-8)\)
(c) \(\log_9 (2x+21)+1=\log_3(x-3)\)
Rešitev: (a) \(x=7\), (b) \(x=5\), (c) \(x=30\) (Namig: Lahko si pomagaš s formulo za prehod na novo osnovo.)(a) \(\displaystyle \frac{\log_3 x + 2}{\log_3 (x+2)}=2\)
(b) \(\displaystyle \frac{\big(2\log x\big)^2}{2\log x^2 -1}=1\)
(c) \(\displaystyle\frac{1+\log_6 (x+3)}{\log_6 (x-9)}=2\)
(d) \(\displaystyle\frac{\log(x-9)}{\,\log(x+2)+2\,} = \frac{1}{\,2\,}\)
Rešitev: (a) \(x_1=1,~ x_2=4\), (b) \(x=\sqrt{10}\), (c) \(x=21\), (d) \(x=119\)(a) \((\log x+1)\log x +1=7\)
(b) \(\log_x 100+\log_{10x} 1000 =2\)
(c) \(\displaystyle\frac{2\log_4 x +1}{2\log_4 (x+6)}=1\)
(d) \(\displaystyle\frac{1}{\,3\,}=\frac{\log(x-2)}{\log(x^3-26)}\)
Rešitev: (a) \(x_1=100,~ x_2=\frac{1}{1000}\), (b) \(x_1=100,~ x_2=\frac{\sqrt{10}}{10}\), (c) \(x=6\), (d) enačba nima rešitve(a) \(\log_4(2x-3)=-\log_4(x-2)\)
(b) \(2 \log x + \log(x-1)= \log(2x^2+4x-12)\)
(c) \(\displaystyle\frac{\log x+1}{\log x+5}=\frac{\log x}{\log x+3}\)
(d) \(\displaystyle\log_8 (x-3)-\log\!\lower0.8em{\textstyle\frac{1}{8}} x=\frac{2}{\,3\,}\)
Rešitev: (a) \(x=\frac{5}{2}\), (b) \(x_1=2,~ x_2=3\), (c) \(x=1000\), (d) \(x=4\)(a) \(\displaystyle 3^{\log100x}-3^{\log10x}+3^{\log x}=63\)
(b) \(\displaystyle\log_2(3^x+7)=\log_2(3^{x+1}+3)+1\)
Rešitev: (a) \(x=100\), (b) \(x=\log_3 \frac{1}{5}\doteq-1,\!465\)