\[\log{\frac{75}{16}}-2\log{\frac{5}{9}}+\log{\frac{32}{243}}=\log{2}\]
1
Use Power Rule: log⁡bxc=clog⁡bx\log_{b}{{x}^{c}}=c\log_{b}{x}.
log⁡7516−log⁡(59)2+log⁡32243=log⁡2\log{\frac{75}{16}}-\log{{(\frac{5}{9})}^{2}}+\log{\frac{32}{243}}=\log{2}

2
Use Division Distributive Property: (xy)a=xaya{(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}.
log⁡7516−log⁡5292+log⁡32243=log⁡2\log{\frac{75}{16}}-\log{\frac{{5}^{2}}{{9}^{2}}}+\log{\frac{32}{243}}=\log{2}

3
Simplify  52{5}^{2}  to  2525.
log⁡7516−log⁡2592+log⁡32243=log⁡2\log{\frac{75}{16}}-\log{\frac{25}{{9}^{2}}}+\log{\frac{32}{243}}=\log{2}

4
Simplify  92{9}^{2}  to  8181.
log⁡7516−log⁡2581+log⁡32243=log⁡2\log{\frac{75}{16}}-\log{\frac{25}{81}}+\log{\frac{32}{243}}=\log{2}

5
Use Product Rule: log⁡b(xy)=log⁡bx+log⁡by\log_{b}{(xy)}=\log_{b}{x}+\log_{b}{y}.
log⁡(75162581×32243)=log⁡2\log{(\frac{\frac{\frac{75}{16}}{25}}{81}\times \frac{32}{243})}=\log{2}

6
Simplify  75162581\frac{\frac{\frac{75}{16}}{25}}{81}  to  7516×25×81\frac{75}{16\times 25\times 81}.
log⁡(7516×25×81×32243)=log⁡2\log{(\frac{75}{16\times 25\times 81}\times \frac{32}{243})}=\log{2}

7
Simplify  16×2516\times 25  to  400400.
log⁡(75400×81×32243)=log⁡2\log{(\frac{75}{400\times 81}\times \frac{32}{243})}=\log{2}

8
Simplify  400×81400\times 81  to  3240032400.
log⁡(7532400×32243)=log⁡2\log{(\frac{75}{32400}\times \frac{32}{243})}=\log{2}

9
Simplify  7532400\frac{75}{32400}  to  1432\frac{1}{432}.
log⁡(1432×32243)=log⁡2\log{(\frac{1}{432}\times \frac{32}{243})}=\log{2}

10
Use this rule: ab×cd=acbd\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}.
log⁡(1×32432×243)=log⁡2\log{(\frac{1\times 32}{432\times 243})}=\log{2}

11
Simplify  1×321\times 32  to  3232.
log⁡(32432×243)=log⁡2\log{(\frac{32}{432\times 243})}=\log{2}

12
Simplify  432×243432\times 243  to  104976104976.
log⁡32104976=log⁡2\log{\frac{32}{104976}}=\log{2}

13
Simplify  32104976\frac{32}{104976}  to  26561\frac{2}{6561}.
log⁡26561=log⁡2\log{\frac{2}{6561}}=\log{2}

14
Since log⁡26561=log⁡2\log{\frac{2}{6561}}=\log{2} is false, there is no solution.
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