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 September 23rd, 2012, 05:45 PM #1 Senior Member   Joined: Apr 2012 Posts: 799 Thanks: 1 exponential problem $(67)^x=27$ $(603)^y=81$ $\text find\,\, :\,\,\frac {3}{x} \,\, - \,\,\frac {4}{y} \text$ $\text Ans:-2\text$
 September 23rd, 2012, 06:18 PM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond Re: exponential problem $^{67\,=\,27^{\small{1/x}}} \\ ^{603\,=\,81^{\small{1/y}}} \\ ^{\frac{603}{67}\,=\,\frac{81^{\small{1/y}}}{27^{\small{1/x}}}} \\ ^{9\,=\,\frac{3^{\small{1/y}}\,\cdot\,27^{\small{1/y}}}{27^{\small{1/x}}}} \\ ^{9\,=\,3^{\small{1/y}}\,\cdot\,27^{\small{1/y-1/x}}} \\ ^{3^{\small{2}}\,=\,3^{\small{1/y}}\,\cdot\,3^{\small{3(1/y-1/x)}}} \\ ^{2\,=\,\frac{3}{y}\,-\,\frac{3}{x}\,+\,\frac{1}{y}} \\ ^{2\,=\,\frac{4}{y}\,-\,\frac{3}{x}} \\ ^{-2\,=\,\frac{3}{x}\,-\,\frac{4}{y}}$
 September 23rd, 2012, 06:38 PM #3 Senior Member   Joined: Apr 2012 Posts: 799 Thanks: 1 Re: exponential problem greg1313:well done
 September 23rd, 2012, 06:45 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: exponential problem $67=3^{\frac{3}{x}}\:\therefore\:\frac{3}{x}=\log_3 (67)$ $603=3^{\frac{4}{y}}\:\therefore\:\frac{4}{y}=\log_ 3(9)+\log_3(67)=2+\log_3(67)$ Hence: $\frac{3}{x}-\frac{4}{y}=-2$
 September 24th, 2012, 03:52 PM #5 Global Moderator   Joined: Dec 2006 Posts: 20,926 Thanks: 2205 Re: exponential problem $3^{\small3/x\,-\,4/y}\,=\,\frac{{\small27}^{1/x}}{{\small81}^{1/y}}\,=\,\frac{\small67}{\small603}\,=\,3^{\small-2},\text{ so }\frac{\small3}{x}\,-\,\frac{\small4}{y}\,=\,-2.$
 September 24th, 2012, 07:36 PM #6 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: exponential problem Again i like skipjack answer, deja vu?
 September 24th, 2012, 08:28 PM #7 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: exponential problem Yeah, skipjack always presents ingeniously creative proofs. I like his methods too.
 September 24th, 2012, 09:13 PM #8 Senior Member   Joined: Apr 2012 Posts: 799 Thanks: 1 Re: exponential problem I agree "skipjack 's solution is very clear and succinct"

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