solve for x

Question

15^9x=13^-x-7

 

write the exact answer using base -10 logarithm

Answer 1

Well I hope so...

because if its not then after adding 7 to both sides the ratio becomes:

 

 

  15^(9x) +7  = 1/ (13^x)

 

 

Cross multiplication gives  [15^(9x) +7]* 13^x = 1  <---  game over

 

So for f(x) = 15^(9x) - 13^(-x) + 7, Desmos shows the root around -3/4.

The bisection method was able to give me 12 digits of accuracy after 44 iterations

on the interval [-0.9, -0.5]

 

X = -0.75865441397782

 

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OTHERWISE, the equation reads 15^(9x) = 13^(-x-7)

 

Taking log of both sides,   9x* log15 = (-x-7) log 13

 

                                      9x log 15 = -log13 X - 7log13

 

                                      9x log 15 + log 13 X = -7log13

 

                                     X ( 9 log 15 + log 13) = -7 log 13

 

                                     X =  (-7log13)/( 9 log 15 + log 13)

 

 

                                  which is approximately X = -0.666532209

 

                                    and it checks.  MS Excel shows zeros all the way down to the

                                    7th digit on both sides when its plugged in