Solve:
a) 1/(2^x) = 1/(x+2)
b) 1/(2^x) > 1/(x^2)
a) To solve the equation 1/(2^x) = 1/(x+2), we can start by cross-multiplying:
(2^x) * (x + 2) = 1
Next, we'll express the equation in exponential form by rewriting 2^x as (2^1)^x = 2^(x*1) = 2^x:
2^x * (x + 2) = 1
Now, distribute the 2^x to both terms within the parentheses:
(x + 2) * 2^x = 1
To simplify the equation further, let's isolate the variable terms by dividing both sides by (x + 2):
2^x = 1/(x + 2)
Now, we need to solve for x using logarithms. Take the logarithm (base 2) of both sides:
log2(2^x) = log2(1/(x + 2))
x * log2(2) = log2(1) - log2(x + 2)
Since log2(2) equals 1 and log2(1) equals 0, the equation becomes:
x = -log2(x + 2)
At this point, we have an equation in terms of x. To find the exact value of x, we need to use numerical methods or graphing calculators. There is no algebraic solution.
b) To solve the inequality 1/(2^x) > 1/(x^2), we can start by cross-multiplying:
(x^2) * (2^x) < 1
Next, express the inequality in exponential form:
2^x * (x^2) < 1
Again, distribute the 2^x to both terms within the parentheses:
(x^2) * 2^x < 1
Now, let's simplify the inequality by dividing both sides by (x^2):
2^x < 1/(x^2)
To further simplify, take the logarithm (base 2) of both sides:
log2(2^x) < log2(1/(x^2))
x * log2(2) < log2(1) - log2(x^2)
Since log2(2) equals 1 and log2(1) equals 0, the inequality becomes:
x < -log2(x^2)
To find the solution for x, we need to simplify the right-hand side. Using logarithmic properties, -log2(x^2) can be rewritten as log2(1/x^2):
x < log2(1/x^2)
Again, there is no algebraic solution for this inequality. To find the exact value of x, numerical methods or graphing calculators are necessary.