NEED ASAP! solve 5^4-x = 7^3x+1. Write answer in exact form, not in decimal approximations for any values.

Question

Solve: 5^4-x = 7^3x+1

answer in exact form, not in decimal

Do not use properties of logarithms to simplify final answer.

Answer 1

 

                           5 ^{4 – x} = 7 ^{3x + 1}

We will take log from left and right parts, then

                       log (5 ^{4 – x}) = log (7 ^{3x + 1})

                  (4 – x ) log 5 = (3 x + 1) log 7

                  (4 – x) / (3x + 1) = log 7 / log 5

Let y = log 7 / log 5, then

                                          (4 – x) / (3x + 1) = y

Or

                                                    4 – x = y (3x + 1)

                                                   4 – x = 3 yx + y

                                            – x – 3 yx = y – 4

                                                        x + 3yx = 4 – y

Factor

                                                      x (1 + 3y) = 4 – y

From here

                                              x = (4 – y) / (1 + 3y)

Finally,

                           x = ( 4 –  log 7 / log 5 ) / ( 1 + 3(log 7 / log 5)