katherinesmith
  • katherinesmith
6^(x - 3) = 52
Mathematics
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SOLVED
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katieb
  • katieb
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katherinesmith
  • katherinesmith
\[6^{x - 3} = 52\]
katherinesmith
  • katherinesmith
i know how to do it when it's just 6^x = 52 but now that they threw in the x - 3 i don't know what to do.
anonymous
  • anonymous
\[6^{x-3}=52\iff x-3=\frac{\ln(52)}{\ln(6)}\] by the change of base formula

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More answers

phi
  • phi
same idea take the log of both sides use the rule \[ \log(a^b) = b \log(a) \]
anonymous
  • anonymous
the \(-3\) is a red herring just add 3 at the end
katherinesmith
  • katherinesmith
do i plug 52 in for a?
phi
  • phi
so the problem (after taking the log of both sides) is (x+3) log(6) = log(52) log(6) is just a number. divide both sides by log(6) (x+3)= log(52)/log(6) or x+3 = log(52)/log(6) add -3 to both sides x= log(52)/log(6) - 3 it is now calculator time if you want to get a number...
katherinesmith
  • katherinesmith
i got a negative number...
phi
  • phi
that is because I made a typo it is x-3 = log(52)/log(6) so x = log(52)/log(6)+3
katherinesmith
  • katherinesmith
i was wondering why you changed signs but i didn't know if that was part of some weird rule.
phi
  • phi
once you get an answer, you can check it in the original equation to make sure it works.
katherinesmith
  • katherinesmith
now if i have \[10^{x + 4} = 1000\] can i do \[10^{x + 4} = 10^{2}\] and then what
phi
  • phi
you mean 10^3 (10^2 is 10*10= 100) if the bases are equal, you can set the exponents equal and solve x+4 = 3 or, if you use logs: log( 10^(x+4) ) = log(10^3) (log is base 10) so this gives you (x+4) log(10) = 3 log(10) log base 10 of 10 is 1, and you get x+4 = 3 which matches the first (and faster) way
katherinesmith
  • katherinesmith
gotcha. okay that is easy.

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