Solve the Equation:
1. 2^x + 2^-x = 5
2. Log2x + log2(4 – x) = 0
thanks. i'm not sure how to approach this at all.
2^x + 1/2^x = 5
let y = 2^x
y + 1/y = 5
y^2 + 1 = 5 y
y^2 -5y +1 = 0
y = [5 +/- sqrt(21) ] /2
y = [2.5 +/- 2.3] approximately
y = 4.8 or y = .2
if 2^x = 4.8
x log 2 = log 4.8 solve for x
if y = .2, same deal,perhaps same answer
Now on this one I am not sure what you mean.
log base 2 (x) or log (2x)
I will assume base 2
then
log x + log (4-x) = 0 where logs are base 2
adding logs is multiplying
log (x(4-x)) = 0
x(4-x) = 2^0 which is one
4x - x^2 = 1
x^2 -4x +1 = 0
solve for x
To solve these equations, we can use algebraic methods. Here's how you can approach each equation:
1. 2^x + 2^(-x) = 5:
To solve this equation, we can use a substitution. Let's say y = 2^x. Then the equation becomes y + 1/y = 5.
Multiply both sides of the equation by y to remove the fraction:
y^2 + 1 = 5y
Rearrange the equation:
y^2 - 5y + 1 = 0
Now we have a quadratic equation. We can solve this quadratic equation using the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -5, and c = 1. Plug in these values to the formula:
y = (5 ± √((-5)^2 - 4(1)(1))) / (2(1))
Simplify the equation:
y = (5 ± √(25 - 4)) / 2
y = (5 ± √21) / 2
So, we have two possible values for y:
y₁ = (5 + √21) / 2
y₂ = (5 - √21) / 2
Now we substitute y back into the equation y = 2^x:
2^x = (5 + √21) / 2
2^x = (5 - √21) / 2
To find the value of x, take the logarithm(base 2) of both sides:
x = log₂((5 + √21) / 2)
x = log₂((5 - √21) / 2)
So the solution for equation 1 is:
x = log₂((5 + √21) / 2) or x = log₂((5 - √21) / 2)
2. log₂x + log₂(4 – x) = 0:
To solve this logarithmic equation, we can use the properties of logarithms.
Using the product rule of logarithms, we can rewrite the equation as:
log₂(x(4 – x)) = 0
Applying the log₂ function to both sides:
x(4 – x) = 2^0
Simplifying the right side:
x(4 – x) = 1
Expanding the left side:
4x - x² = 1
Rearranging the terms:
x² - 4x + 1 = 0
Now we have a quadratic equation. We can solve it using the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 1, b = -4, and c = 1. Plug in these values into the formula:
x = (4 ± √((-4)² - 4(1)(1))) / (2(1))
Simplify the equation:
x = (4 ± √(16 - 4)) / 2
x = (4 ± √12) / 2
So, we have two possible values for x:
x₁ = (4 + √12) / 2
x₂ = (4 - √12) / 2
Simplifying further:
x₁ = 2 + √3
x₂ = 2 - √3
So the solution for equation 2 is:
x = 2 + √3 or x = 2 - √3