(3^x)+(5^{x+3})=(3^{x+4})-(1/3)*( 5^{x+2}).
Solve
3^x + 5^(x+3) - 3^(x+4) + (1/3)5^(x+2)= 0
3^x(1 - 3^4) + 5^(x+2)(5 + (1/3)(1) ) = 0
-80(3^x) + (16/3)(5^(x+2) = 0
times -3/16
15(3^x) - 5^(x+2) = 0
15(3^x) = 5^(x+2)
divide by 5
3(3^x) = 5^(x+1)
3^(x+1) = 5^(x+1)
this can only be true if x = -1
that is, if 3^0 = 5^0 , which is true
3^x + 5^(x+3) - 3^(x+4) + (1/3)5^(x+2)= 0
3^x(1 - 3^4) + 5^(x+2)(5 + (1/3)(1) ) = 0
-80(3^x) + (16/3)(5^(x+2) = 0
times -3/16
15(3^x) - 5^(x+2) = 0
15(3^x) = 5^(x+2)
divide by 5
3(3^x) = 5^(x+1)
3^(x+1) = 5^(x+1)
this can only be true if x = -1
that is, if 3^0 = 5^0 , which is true
just start by combining like powers:
(3^x)+(5^{x+3})=(3^{x+4})-(1/3)*( 5^{x+2})
3^x + 125*5^x = 81*3^x - 3^-1 * 25*5^x
3*3^x + 375*5^x = 243*3^x - 25*5^x
(375+25)*5^x = (243-3)*3^x
400*5^x = 240*3^x
5*5^x = 3*3^x
5^(x+1) = 3^(x+1)
(x+1)log5 = (x+1)log3
(x+1)(log5-log3) = 0
x = -1
To solve the equation (3^x) + (5^(x+3)) = (3^(x+4)) - (1/3)*(5^(x+2)), we need to simplify the equation and isolate the variable x. Here's how you can do it step by step:
Step 1: Expand the exponents using the exponentiation properties.
- (3^x) + (5^x)*(5^3) = (3^x)*(3^4) - (1/3)*(5^x)*(5^2)
Step 2: Simplify the equation by combining like terms.
- (3^x) + (125)*(5^x) = (81)*(3^x) - (1/3)*(25)*(5^x)
Step 3: Move all terms with x to one side of the equation and the constant terms to the other side.
- (3^x) - (81)*(3^x) = -(125)*(5^x) + (1/3)*(25)*(5^x)
Step 4: Combine like terms on each side of the equation.
- -(80)*(3^x) = -(100)*(5^x)
Step 5: Divide both sides of the equation by a common factor to simplify.
- (80/100)*(3^x) = (5^x)
Step 6: Simplify the coefficients.
- (4/5)*(3^x) = (5^x)
Step 7: Divide both sides of the equation by (3^x) to isolate x.
- (4/5) = (5^x)/(3^x)
Step 8: Rewrite the right side of the equation using the properties of exponents.
- (4/5) = (5/3)^x
Step 9: Take the logarithm of both sides of the equation.
- log((4/5)) = log((5/3)^x)
Step 10: Apply the power rule of logarithms to simplify the equation.
- log((4/5)) = x*log((5/3))
Step 11: Divide both sides of the equation by log((5/3)) to solve for x.
- x = log((4/5))/log((5/3))
You can use a calculator or a software tool to compute the value of x.