document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=337518>Question 337518</A>: <I><SPAN STYLE=\"word-break: break-all;\"> PH^t=Vk^t<BR>\n" );
document.write( "I need to use logarithms to solve for t, however I am not sure what to do.<BR>\n" );
document.write( "tlog(Ph)=tlog(Vk) or<BR>\n" );
document.write( "log(p)*tlog(h)=log(v)*tlog(k)<BR>\n" );
document.write( "I am not sure which one is right and where to go on from here. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #241974 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=241974\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If there were parentheses like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (PH)^t\ =\ (Vk)^t\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then your first attempt would have been correct.  As it stands though, neither is correct.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ PH^t\ =\ Vk^t\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Taking the log of both sides was absolutely the correct thing to do.  Personally, I would have taken the natural log rather than the base 10 log, but it makes no difference in the final analysis.  You say to-may-to, I say to-mah-to.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log(PH^t)\ =\ \log(Vk^t)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Next use\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x) + \log_b(y) = \log_b(xy)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to write:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log(P)\ +\ \log(H^t)\ =\ \log(V)\ +\ \log(k^t)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then use:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x^n)\ =\ n\log_b(x)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to write (and here is where we start to deviate -- note the plus signs in mine as compared to your multiplication signs.  Of course, multiplying by adding is really the point of logarithms, now isn't it?):\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log(P)\ +\ t\log(H)\ =\ \log(V)\ +\ t\log(k)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Collect the terms with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\"> on the left and the other two terms on the right:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t\log(H)\ -\ t\log(k)\ =\ \log(V)\ -\ \log(P)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factor out <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\">:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t\left(\log(H)\ -\ \log(k)\right)\ =\ \log(V)\ -\ \log(P)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Next use:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x) - \log_b(y) = log_b\left(\frac{x}{y}\right)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to write:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t\log\left(\frac{H}{k}\right)\ =\ \log\left(\frac{V}{P}\right)\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log\left(\frac{H}{k}\right)\">:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ \frac{\log\left(\frac{V}{P}\right)}{\log\left(\frac{H}{k}\right)}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And that, as they say, is that.  And just as an \"oh by the way,\" the following identical relationship is true:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \frac{\log\left(\frac{V}{P}\right)}{\log\left(\frac{H}{k}\right)}\ \equiv\ \frac{\ln\left(\frac{V}{P}\right)}{\ln\left(\frac{H}{k}\right)}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://www.evolvefish.com/fish/media/E-FlyingSpaghettiEmblem.gif\"><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );