solve the following exponential equation.
(e^x-e^-x)/2=6
what is the exact answer?
x=?
what is the answer approx to 3 decimal places?
x=?
To solve the exponential equation (e^x - e^-x)/2 = 6, we can follow these steps:
Step 1: Multiply both sides of the equation by 2 to eliminate the fraction, giving us:
(e^x - e^-x) = 12
Step 2: Multiply both sides of the equation by e^x to get rid of the negative exponential term:
e^x(e^x - e^-x) = 12e^x
Step 3: Distribute the e^x term:
(e^(2x) - 1) = 12e^x
Step 4: Move all terms to one side of the equation:
e^(2x) - 12e^x - 1 = 0
Now we have a quadratic equation in terms of e^x. Let's define u as e^x and rewrite the equation:
u^2 - 12u - 1 = 0
Step 5: Solve the quadratic equation. We can use the quadratic formula:
u = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = 1, b = -12, and c = -1. Plugging these values into the quadratic formula, we get:
u = (-(-12) ± √((-12)^2 - 4(1)(-1)))/(2(1))
Simplifying further:
u = (12 ± √(144 + 4))/2
u = (12 ± √148)/2
u = (12 ± √(74 * 2))/2
u = (12 ± 2√74)/2
u = 6 ± √74
Step 6: Convert u back to e^x:
For u = 6 + √74, we have:
e^x = 6 + √74
Taking natural logarithm on both sides:
x = ln(6 + √74)
For u = 6 - √74, we have:
e^x = 6 - √74
x = ln(6 - √74)
The exact answers for x are x = ln(6 + √74) and x = ln(6 - √74).
To get the approximate answer to 3 decimal places, we can use a calculator or software to evaluate ln(6 + √74) and ln(6 - √74). The approximate values will be:
x ≈ 2.341
x ≈ -2.341