# 9^(3x-2)- 7^(6x-4)

Posted 9 years ago
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 Seams that 'step by step' solution for this eqation (for real and imaginary roots)are unavailable in wolfram alpha... Why ?And why are some transcedental functions are 'overkill' for wolfram alpha ?
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Posted 9 years ago
 Thnx guys... i really appreciate your help
Posted 9 years ago
 In[8]:= Solve[9^(3 x - 2) == 7^(6 x - 4), x] // FullSimplifyDuring evaluation of In[8]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>Out[8]= {{x -> 2/3}}
Posted 9 years ago
 However, if you move one term to the other side of the equal sign, then you can get a step-by-step solution.Solve for x:9^(3 x-2) = 7^(6 x-4)Look for an exponent to divide both sides by.Divide both sides by 9^(3 x-2):1 = 7^(6 x-4) 9^(2-3 x)Reverse the equality in 1 = 7^(6 x-4) 9^(2-3 x) in order to isolate x to the left hand side.1 = 7^(6 x-4) 9^(2-3 x) is equivalent to 7^(6 x-4) 9^(2-3 x) = 1:7^(6 x-4) 9^(2-3 x) = 1Write exponents in 7^(6 x-4) 9^(2-3 x) = 1 in terms of a common base.7^(6 x-4) 9^(2-3 x) = e^(log(7^(6 x-4))) e^(log(9^(2-3 x))) = e^((6 x-4) log(7)) e^((2-3 x) log(9)) = exp((6 x-4) log(7)+(2-3 x) log(9)):exp(log(7) (6 x-4)+log(9) (2-3 x)) = 1Eliminate the exponential from the left hand side.Take the natural logarithm of both sides:log(7) (6 x-4)+log(9) (2-3 x) = (2 i) pi n  for  n element ZWrite the linear polynomial on the left hand side in standard form.Expand and collect in terms of x:(6 log(7)-3 log(9)) x-4 log(7)+2 log(9) = (2 i) pi n  for  n element ZIsolate terms with x to the left hand side.Subtract 2 log(9)-4 log(7) from both sides:(6 log(7)-3 log(9)) x = 4 log(7)-2 log(9)+(2 i) pi n  for  n element ZSolve for x.Divide both sides by 6 log(7)-3 log(9):Answer: |  | x = (4 log(7))/(6 log(7)-3 log(9))-(2 log(9))/(6 log(7)-3 log(9))+((2 i) pi n)/(6 log(7)-3 log(9))  for  n element Z
Posted 9 years ago
 Thanks mr. Frank Kampas... ;)
Posted 9 years ago
 In[18]:= Reduce[9^(3 x - 2) == 7^(6 x - 4), x]Out[18]= C[1] \[Element] Integers && x == -((2 I \[Pi] C[1] - 4 Log[3] + 4 Log[7])/(6 (Log[3] - Log[7])))In fact it is 9^y==7^(2 y) which of course you see is (3^2)^y == 7^(2 y) which is 3^(2 y)==7^(2 y) which is 3^z ==7^z you can do by hand only ...