Final answer:
The solution of the equation 6eˣ - 4e⁻ˣ = 5 is x = ln(8) - ln(3).
None of the given options is correct
Step-by-step explanation:
To solve the equation 6eˣ - 4e⁻ˣ = 5 for x, we can use a substitution. Let's substitute eˣ with a variable, let's say y. Then the equation becomes:
6y - 4/y = 5.
To solve this equation, we can multiply through by y to get rid of the fraction:
6y² - 4 = 5y.
Rearranging the equation, we have:
6y² - 5y - 4 = 0.
To factor this quadratic equation, we can look for two numbers that multiply to give -24 (product of the coefficients 6 and -4) and add up to -5 (coefficient of the middle term). Those numbers are -8 and 3. So, we can factor the equation as:
(2y + 3)(3y - 8) = 0.
Setting each factor equal to zero, we have:
2y + 3 = 0 or 3y - 8 = 0.
Solving these two equations, we find:
2y = -3 or 3y = 8.
Dividing both sides of the first equation by 2, we get:
y = -3/2.
And dividing both sides of the second equation by 3, we get:
y = 8/3.
Now, we need to find the values of x, not y. To find x, we substitute the values of y back into the equation eˣ = y.
For y = -3/2:
eˣ = -3/2.
To solve for x, we can take the natural logarithm of both sides:
x = ln(-3/2).
However, the natural logarithm is undefined for negative numbers, so this solution is not valid.
For y = 8/3:
eˣ = 8/3.
Taking the natural logarithm of both sides:
x = ln(8/3).
Simplifying the expression, we get:
x = ln(8) - ln(3).
So, the correct answer is: x = ln(8) - ln(3).
Therefore, the correct answer is not provided in the given options A, B, C, or D.