a)

Rozwiązujmy równanie:

$5^x=6^{x+1}$  

$5^x=6^x\cdot 6^1|:6^x$

${\left(\frac{5}{6}\right)}^x=6$

${\log }_{\frac{5}{6}}{\left(\frac{5}{6}\right)}^x={\log }_{\frac{5}{6}}6$

$x={\log }_{\frac{5}{6}}6$

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b)

Rozwiązujmy równanie:

$11^{x-3}=13^{3-x}$

$11^x\cdot 11^{-3}=13^3\cdot 13^{-x}$

$11^x\cdot {\left(\frac{1}{11}\right)}^3=13^3\cdot {\left(\frac{1}{13}\right)}^x|:{\left(\frac{1}{13}\right)}^x$

$11^x:{\left(\frac{1}{13}\right)}^x\cdot {\left(\frac{1}{11}\right)}^3=13^3|:{\left(\frac{1}{11}\right)}^3$

$11^x:{\left(\frac{1}{13}\right)}^x=13^3:{\left(\frac{1}{11}\right)}^3$

${\left(11:\frac{1}{13}\right)}^x={\left(13:\frac{1}{11}\right)}^3$

${\left(11\cdot 13\right)}^x={\left(13\cdot 11\right)}^3$

$143^x=143^3$

$x=3$

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c)

Rozwiązujmy równanie:

$7^{x+1}=3^{2x-2}$

$7^x\cdot 7=3^{2x}\cdot 3^{-2}$

$7^x\cdot 7=9^x\cdot \frac{1}{9}$

${\left(\frac{7}{9}\right)}^x=\frac{1}{9}:7$

${\left(\frac{7}{9}\right)}^x=\frac{1}{9}\cdot \frac{1}{7}$

${\left(\frac{7}{9}\right)}^x=\frac{1}{63}$

${\left(\frac{9}{7}\right)}^x=63$  

${\log }_{\frac{9}{7}}{\left(\frac{9}{7}\right)}^x={\log }_{\frac{9}{7}}63$

$x={\log }_{\frac{9}{7}}63$

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d)

Rozwiązujmy równanie:

$3^{2x}=2^x\cdot 4^{x-1}$

$9^x=2^x\cdot 4^x\cdot 4^{-1}$

$9^x=8^x\cdot \frac{1}{4}$

${\left(\frac{9}{8}\right)}^x=\frac{1}{4}$

${\log }_{\frac{9}{8}}{\left(\frac{9}{8}\right)}^x={\log }_{\frac{9}{8}}\frac{1}{4}$

$x={\log }_{\frac{9}{8}}\frac{1}{4}$

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e)

Rozwiązujmy równanie:

$4\cdot 2^{x-3}=10^x\cdot 3^{x-1}$

$4\cdot 2^x\cdot 2^{-3}=10^x\cdot 3^x\cdot 3^{-1}$

$4\cdot 2^x\cdot \frac{1}{8}=10^x\cdot 3^x\cdot \frac{1}{3}$

$2^x\cdot \frac{1}{2}=30^x\cdot \frac{1}{3}$

${\left(\frac{2}{30}\right)}^x=\frac{2}{3}$

${\left(\frac{1}{15}\right)}^x=\frac{2}{3}$

$15^x=\frac{3}{2}$

${\log }_{15}15^x={\log }_{15}\frac{3}{2}$

$x={\log }_{15}\frac{3}{2}$