EXPONENTIAL EQUATION
Find the X
3^x - 82/9 = -3^-x
3^x + 3^-x = 82/9
3^x + 1/3^x = 82/9
(3^2x + 1)/3^x = 82/9
clearly, x=2, but let's go on.
9(3^2x + 1) = 82*3^x
9*3^2x - 82*3^x + 9 = 0
think of this as 9u^2-82u+9, and you get
(9*3^x - 1)(3^x - 9) = 0
3^x = 9 or 1/9
x = 2 or -2
aha. an unexpected solution!
Thank you so much. It really helps me 😂
To find the value of x in the exponential equation 3^x - 82/9 = -3^-x, we can use algebraic manipulations and solve for x step by step. Here's how you can go about it:
Step 1: Simplify the equation by multiplying both sides by 3^x to remove the negative exponent:
(3^x)(3^x) - (82/9)(3^x) = -1
Step 2: Apply the rules of exponents by combining the terms on the left side of the equation:
3^(2x) - (82/9)(3^x) = -1
Step 3: Rewrite the equation using a common base:
(3^x)^2 - (82/9)(3^x) = -1
Step 4: Let's make a substitution for ease of solving. Let y = 3^x. By substituting y into the equation, it becomes:
y^2 - (82/9)y = -1
Step 5: Rearrange the equation to bring all terms to one side:
y^2 - (82/9)y + 1 = 0
Step 6: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, it is not easy to factor, so we'll use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 1, b = -(82/9), and c = 1. Plug in these values:
y = (-(82/9) ± √((-(82/9))^2 - 4(1)(1))) / (2(1))
Step 7: Simplify the equation:
y = (-(82/9) ± √(6724/81 - 4/1)) / 2
Step 8: Further simplify and evaluate the square root:
y = (-(82/9) ± √(6724/81 - 324/81)) / 2
= (-(82/9) ± √(6400/81)) / 2
= (-(82/9) ± (80/9)) / 2
Step 9: Break it down into two separate possibilities:
a) y = (-(82/9) + (80/9)) / 2
b) y = (-(82/9) - (80/9)) / 2
Step 10: Solve for y:
a) y = (-2/9) / 2
= -1/9
b) y = (-162/9) / 2
= -9
Step 11: Substitute back y = 3^x to find x:
a) -1/9 = 3^x
Taking the logarithm base 3 of both sides:
log₃(-1/9) = x
b) -9 = 3^x
Taking the logarithm base 3 of both sides:
log₃(-9) = x
Step 12: Evaluate the logarithm using a calculator:
a) log₃(-1/9) ≈ -2/3
b) log₃(-9) is undefined. This means that there is no real value of x that satisfies this equation.
Therefore, the possible solution for x is approximately -2/3.