Question 1016251
7^(2x) = 49^(3x+1)


one way to solve this is:


start with:


7^(2x) = 49^(3x+1)


since 49 = 7^2, this equation becomes:


7^(2x) = (7^2)^(3x+1)


this becomes 7^(2x) = 7^(2*(3x+1)) which becomes:


7^(2x) = 7^(6x + 2)


since the base is the same, you can just set the exponents equal to each other and solve for x.


you get 2x = 6x + 2


solve for x to get x = -1/2


another way to solve it is:


start with:


7^(2x) = 49^(3x+1)


take the log of both sides of the equation to get:


log(7^(2x) = log(49^(3x+1))


this becomes 2x * log(7) = (3x+1) * log(49)


since 49 is equal to 7^2, this becomes 2x * log(7) = (3x+1) * log(7^2).


this becomes 2x * log(7) = 2 * (3x + 1) * log(7) which becomes:


2x * log(7) = (6x + 2) * log(7)


if  you divide both sides of this equation by log(7), you get:


2x = 6x + 2


solve for x to get x = -1/2


if you did not recognize that 49 = 7^2 and that you could then get 2 * log(7), you could still have solved as follows:


start with:


2x * log(7) = (3x+1) * log(49)


find log(7) and find log(49) and the equation becomes:


2x * .84509804 = (3x + 1) * 1.69019608


simplify to get:


1.69019608 * x = 3x * 1.69019608 + 1 * 1.69019608


this becomes 1.69019608 * x = 5.07058824 * x + 1.69019608


subtract 1.69019608 * x from both sides of the equation and subtract 1.69019608 from both sides of the equation to get:


-1.69019608 = 5.07058824 * x - 1.69019608 * x


combine like terms to get -1.69019608 = 3.38039216 * x


divide both sides of this equation by 3.38039216 and solve for x to get:


x = 3.38039216 / -1.69019608 = -.5 which is the same as -1/2.


it was a lot messier because of the arithmetic involved but you go the same answer.