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Solve for x in the equation: 6e^x - 4e^-x = 5.

A) x = ln(2)
B) x = -ln(2)
C) x = -ln(3)
D) x = ln(3)

User Itsliamoco
by
8.4k points

1 Answer

1 vote

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.

User Chelsie
by
7.7k points
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