Solve the exponential equation by converting to a common base:

Question

Show step by step work:

49^3x-45⁄ 7^4x+1=7×49^-3x+14

Answer 1

49^3x-45⁄ 7^4x+1 = 7×49^-3x+14

First, the thing to note is that 49 = 7 x 7 = 7². So let's replace the 49s with 7²:

49^3x-45⁄ 7^4x+1 = 7·49^-3x+14

(7²)^3x-45⁄ 7^4x+1 = 7·(7²)^-3x+14

Second, use the exponent rule (a^b)^c = a^bc:

7^6x-90⁄ 7^4x+1 = 7·7^-6x+28

OK now we have a common base, 7. The third step is to use the exponent rules a^b/a^c = a^b-c and

a^b·a^c = a^b+c. Remember that 7 = 7¹:

7^{6x-90 - (4x+1)} = 7^1+-6x+28

7^2x-91 = 7^-6x+29

Last step, since the bases are equal, the exponents must be equal:

2x - 91 = -6x + 29

Solve for x.