qustion(1)x^x=100, find x
question(2):2x+2y<4,3x+4y>6 find the value of x&y that satisfy the inequalities
qustion(3):x^x=100^x find x
(1) no algebraic way to do it. Check the graph
(2)
2x+2y < 4
3x+4y > 6
Just plot the lines and shade the areas that intersect
(3)
clearly x=100
3. X^x = 100^x.
x*Logx = x*Log100
Divide both sides by x:
Log X = Log100 = 2.
10^2 = X.
X = 100.
To solve each of these equations, we will need to use a specific approach.
Question (1): x^x = 100
Step 1: Take the logarithm of both sides of the equation. Since x is the base and exponent, we will use logarithms to eliminate the exponent.
log(x^x) = log(100)
Step 2: Use the logarithm properties to simplify the equation. The logarithm of a power is equal to the exponent multiplied by the logarithm of the base.
x * log(x) = log(100)
Step 3: Evaluate the logarithm of 100. The common logarithm (logarithm base 10) will be used in this case.
x * log(x) = 2
Step 4: Divide both sides of the equation by log(x) to isolate the x.
x = 2 / log(x)
Step 5: Now, you can use numerical methods or approximation techniques, such as using a calculator or software, to find the value of x that satisfies the equation.
Question (2): 2x + 2y < 4 and 3x + 4y > 6
To find the values of x and y that satisfy these inequalities, we will follow these steps:
Step 1: Solve the first inequality, 2x + 2y < 4, for x:
2x < 4 - 2y
x < (4 - 2y) / 2
x < 2 - y
Step 2: Solve the second inequality, 3x + 4y > 6, for x:
3x > 6 - 4y
x > (6 - 4y) / 3
x > 2 - (4/3)y
Step 3: Combining the results from Step 1 and Step 2, we have:
2 - y < x < 2 - (4/3)y
Step 4: Now, we can find the range of values for y by considering various intervals and inequalities. For example, if we assume that y ≤ 0, the final inequality will be:
2 ≤ x ≤ 2
So, if y ≤ 0, then any value of x between 2 and 2 will satisfy the inequalities.
For other intervals of y, the range of values for x will vary accordingly.
Question (3): x^x = 100^x
To solve this equation, we will follow these steps:
Step 1: Take the logarithm of both sides of the equation. This will help us eliminate the exponent.
log(x^x) = log(100^x)
Step 2: Apply the logarithm properties to simplify the equation.
x * log(x) = x * log(100)
Step 3: Cancel out the common factor of x.
log(x) = log(100)
Step 4: Evaluate the logarithm of 100.
log(x) = 2
Step 5: Apply the inverse function of logarithm, which is exponentiation. In this case, we need to exponentiate both sides with a base of 10.
10^(log(x)) = 10^2
Step 6: Simplify the equation.
x = 100
The solution to the equation x^x = 100^x is x = 100.