(a) \(f(x)=3-3^x\)
(b) \(f(x)=3^{3-x}\)
(c) \(f(x)=e^{-x}-3\)
(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) \(\cdots=3^{x+2}\)(a) \({\displaystyle 2^{x-1}=\frac{4}{\sqrt{2}}}\)
(b) \(3^{x+2}\cdot \sqrt{3}^{~x}\cdot \Big(\frac{\textstyle 1}{\textstyle 9}\Big)^{x-2}=1\)
(c) \({\displaystyle 3^x=5^{x-1}\cdot \frac{27}{25}}\)
Rešitev: (a) \(x=\frac{5}{2}\), (b) \(x=12\), (c) \(x=3\)(a) \(7^{x+2}+7^x=2450\)
(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) \(9^{2x}-2\cdot 9^x=3\)
(b) \({\displaystyle 3^x-6=\frac{27}{3^x}}\)
(c) \(2^x-2^{5-x}=4\)
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) \(\log_3 81\)
(b) \(\log_4 32\)
(c) \(\log\!\raise-0.4em{\scriptstyle\sqrt{5}}~ \Big(\frac{\textstyle 1}{\textstyle 5\sqrt{5}}\Big)\)
(d) \(\log\!\raise-0.9em{\textstyle\frac{1}{8}}~ \sqrt[\scriptstyle 3]{2}\)
Rešitev: (a) \(\cdots=4\), (b) \(\cdots=\frac{5}{2}\), (c) \(\cdots=-3\), (d) \(\cdots=-\frac{1}{9}\)(a) \(5^{x-1}=\frac{\textstyle 7}{\textstyle 33}\)
(b) \(3^{2x+1}=1234\)
(c) \(\Big(\frac{\textstyle 2}{\textstyle 3}\Big)^{3x}=\frac{\textstyle 125}{\textstyle 9}\)
(d) \(e^{x^2-1}=100\)
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) \(3^{x+1}+3^{x-1}=333\)
(b) \(5^x=13\cdot2^x\)
(c) \(7\cdot 4^{x-2}=5\cdot 3^{x+1}\)
Rešitev: (a) \(x\doteq4,\!1909\), (b) \(x\doteq2,\!7993\), (c) \(x\doteq12,\!2869\)(a) \(f(x)=\log_2(x+4)-2\)
(b) \(f(x)=2-\log_3(-x)\)
(c) \(f(x)=\log\!\raise-0.9em{\textstyle\frac{1}{2}}~ (2x+6)\)
(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) \(\log_2 (a^2 b \sqrt{2})\)
(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) \({\displaystyle \log a-\frac{2\log b+3\log c}{6}}\)
(b) \(3-\log_3 a+\frac{1}{3}\log_3 b-2\log_3 c\)
Rešitev: (a) \(\cdots=\log\frac{a}{\sqrt[6]{b^2c^3}}\), (b) \(\cdots=\log_3 \frac{27\sqrt[3]{b}}{ac^2}\)(a) \(\log_3 x=\frac{1}{2}+\log_3 a+\log_3 b-\log_3 (a+b)\)
(b) \(\ln x=5\ln 2+\ln(a^2-b^2)-3\ln 4-\ln (a-b)\)
Rešitev: (a) \(x=\frac{ab\sqrt{3}}{a+b}\), (b) \(x=\frac{a+b}{2}\)(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) \(x=5\)(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) \(\log_8 \big(x-\frac{1}{3}\big)=\log_8 x-\frac{1}{3}\)
Rešitev: (a) \(x=6\), (b) \(x=9\), (c) \(x=\frac{2}{3}\)(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) \({\displaystyle\frac{\log (x-5)}{\log(7-x)}=\frac{1}{2}}\)
Rešitev: (a) \(x=11\), (b) \(x_1=9,~ x_2=4\), (c) enačba nima rešitve.(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: Pri vseh treh enačbah si lahko pomagaš z uvedbo nove neznanke.)(a) \(\log_4(x-3)=\log_{16}(2x+2)\)
(b) \(\log_2(x-2)=\log_8(7x-8)\)
Rešitev: (a) \(x=7\), (b) \(x=5\)