please solve the equation for 10 pts need answer soon?

Question

49^x+5=343^x-2

Answer 1

INTERPRETATION 1:

Is it 49^(x + 5) = 343^(x - 2)?

First put it in terms of powers of 7:
(7^2)^(x + 5) = (7^3)^(x - 2)

Remember the rule of exponents:
(a^b)^c = a^(bc)

7^2(x + 5) = 7^3(x - 2)

Now take log base 7 of both sides:
2(x + 5) = 3(x - 2)

Expand out:
2x + 10 = 3x - 6

Now solve for x:
x = 10 + 6
x = 16

INTERPRETATION 2:

49^x + 5 = 343^x - 2?
7 = (7^3)^x - (7^2)^x
7^1 = 7^(3x) - 7^(2x)

Take the log base 7 of both sides:
1 = 3x - 2x
1 = x
x = 1

Answer 2

If you mean 49^(x+5) = 343^(x-2) then the answer will be trivial:
take, in the above case, logbase10 both sides:
log10[49^(x+5)] = log10[343^(x-2)]
(x+4)log10[49] = (x-2)log10[343] which is an algebraic equation and easily solved for x.

On the other hand, if you mean the above as written, then:
log10[49^x] = log10[343^x - 2], which, having done this before, is a very difficult and time-consuming. There are two way to solve it the first being solve the RHS for x in terms of log10 with x maintained on the other side. Use a graphical method to solve it by graphing both sides and see where they cross.
The other way is to expand the log10 in a series of both sides and you will get an estimate of x.
If there is another way to solve it, I am really, really interested.

Answer 3

There are indeed two interpretations for this equation.
The one you are very probably looking for has already been solved sufficiently by other answerers and is x=16

The equation as written however, using standard mathematical interpretation (pemdas), is
49^x + 5 = 343^x - 2, and not 49^(x+5) = 343^(x-2).

In the slim chance that is what you are looking for, the (real number) solution is EXACTLY:

x = log{³√[191/54 + (1/27)√(9119.25)]
+ ³√[191/54 - (1/27)√(9119.25)] + 1/3} / log(7)

or x ≈ 0.430449658

The proof of this is lengthy, and as you are probably not looking for this solution anyway, I will just sketch it for any interested parties.

49^x+5=343^x-2
7^(2x) + 5 = 7^(3x) - 2
7^(3x) - 7^(2x) - 7 = 0
Let y = 7^x
Then,
y^3 - y^2 - 7 = 0
this cubic has one real solution and two complex conjugate solutions. Therefore, you can use Cardano's formua to find the exact values for y.
Then, x = log(y)/log(7), shown above.

Answer 4

x = 1

Answer 5

49^x-343^x=-2-5
7^2x-7^3x=-7^1
equating the powers
2x-3x=-1
-x=-1
x=1

Answer 6

7^(2x+10)=7^(3x-6)
2x+10=3x-6
x=16