Question

4e²ˣ-13eˣ+9=0 2⁻⁴ˣ⁺¹=3²ˣ⁻³ log(x+1)+log(x-3)=log(6x²-6) Solve each equation. Round to the nearest hundr 2⁹ˣ=1210 4³ˣ=10

Answer 1

To solve the equation 4e²ˣ-13eˣ+9=0, substitute eˣ with a new variable, y. Use the quadratic formula to solve for y, which results in two solutions: y = 2 and y = 1. Substituting the initial value of y as eˣ and taking the natural logarithm of both sides, we find the solution x = ln(2).

Step-by-step explanation:

To solve the equation 4e²ˣ-13eˣ+9=0, let's substitute eˣ with a new variable, let's say y. So the equation becomes 4y²-13y+9=0. This is a quadratic equation, and we can solve it by factoring or using the quadratic formula. However, this equation cannot be factored easily, so let's use the quadratic formula:

y = (-b ± √(b²-4ac))/(2a)

Plugging in the values from our equation, we have:

y = (-(-13) ± √((-13)²-4(4)(9)))/(2(4))

Simplifying further, we get:

y = (13 ± √(169-144))/(8)

y = (13 ± √(25))/(8)

y = (13 ± 5)/(8)

Thus, solving for y, we have two solutions:

y = (13 + 5)/(8) = 2

y = (13 - 5)/(8) = 1

Now, we substitute the initial value of y as eˣ:

eˣ = 2

To solve for x, we take the natural logarithm (ln) of both sides:

ln(eˣ) = ln(2)

x = ln(2)

So, the solution to the equation 4e²ˣ-13eˣ+9=0 is x = ln(2).