Question

Log1/3(x^2+3)=-cos Pix/2

Answer 1

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Answer 2

To solve the equation $\log_{1/3}(x^2 + 3) = -\cos\left(\frac{\pi x}{2}\right)$, you can follow these steps:

Step 1: Rewrite the equation in exponential form: $x^2 + 3 = 3^{-\cos\left(\frac{\pi x}{2}\right)}$

Step 2: Since $\cos\left(\frac{\pi x}{2}\right)$ can take values between -1 and 1, the right-hand side can be simplified: $3^{-\cos\left(\frac{\pi x}{2}\right)} \leq 3^1 = 3$

Now, we have:

$x^2 + 3 \leq 3$

Step 3: Subtract 3 from both sides of the inequality:

$x^2 \leq 0$

Step 4: Take the square root of both sides. Remember that when you take the square root of a non-negative number (as we have here), there are two possibilities: $x$ can be 0 or $x$ can be any negative number.

$x \leq 0$

So, the solution to the equation $\log_{1/3}(x^2 + 3) = -\cos\left(\frac{\pi x}{2}\right)$ is $x \leq 0$.

Please note that the equation is satisfied for all values of $x$ less than or equal to 0, and there is no unique numerical solution since it is a non-linear equation.