Question

2^(x^2)*7^(x-2)=>16 решите неравенство

Answer 1

Прологарифмируем обе части неравенства по основанию 2:

$\log_2(2^{x^2}\cdot7^{x-2})\geq \log_216\\ \log_2(2^{x^2})+\log_2(7^{x-2})\geq 4\\ x^2+(x-2)\log_27-4\geq 0\\ x^2+\log_27\cdot x+(-2\log_72-4)\geq 0$

Получилось квадратное неравенство, которое можно решить методом интервалов.

Считаем дискриминант трехчлена и его корни:

$D=\log_2^27-4(-2\log_27-4)=\log_2^27+8\log_27+16=(\log_27+4)^2\\\\x_{1,2}=\dfrac{-\log_27\б(\log_27+4)}{2}\\ x_1=-\log_27-2=-\log_228;\ x_2=2\\ (x+\log_228)(x-2)\geq 0$

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       $-\log_228$          2

$x\in (-\infty;\ -\log_228]\cup[2;\ +\infty)$

Ответ: $(-\infty;\ -\log_228]\cup[2;\ +\infty)$

Answer 2

To solve the inequality $2^{x^2} \cdot 7^{x-2} > 16$, we can follow these steps:

Step 1: Rewrite 16 as a power of 2 and 7 to make the equation easier to work with. Step 2: Use logarithms to solve for $x^2$ in terms of $x$. Step 3: Solve the resulting inequality for $x$.

Let's go through each step:

Step 1: Rewrite 16 as a power of 2 and 7: $16 = 2^4$

Step 2: Use logarithms to solve for $x^2$ in terms of $x$: Taking the logarithm of both sides (using any base, but let's use natural logarithm for simplicity): $\ln(2^{x^2} \cdot 7^{x-2}) > \ln(2^4)$

Using the properties of logarithms, we can simplify the left side: $\ln(2^{x^2}) + \ln(7^{x-2}) > \ln(2^4)$

Apply the power rule of logarithms: $x^2 \ln(2) + (x-2) \ln(7) > 4 \ln(2)$

Step 3: Solve the resulting inequality for $x$: Now we have a quadratic-like inequality in terms of $x$. Let's proceed to solve it:

$x^2 \ln(2) + x \ln(7) - 2 \ln(7) > 4 \ln(2)$

To make it easier, we can divide the entire inequality by $\ln(2)$ (since $\ln(2)$ is positive): $x^2 + \frac{x}{\ln(2)} \ln(7) - \frac{2}{\ln(2)} \ln(7) > \frac{4}{\ln(2)}$

Now, we have a quadratic inequality of the form $ax^2 + bx + c > 0$, where $a = 1$, $b = \frac{\ln(7)}{\ln(2)}$, and $c = -\frac{2\ln(7)}{\ln(2)} - \frac{4}{\ln(2)}$.

We can either solve this inequality algebraically or use graphical methods. I will provide the graphical solution:

Graph the function $f(x) = x^2 + \frac{x}{\ln(2)} \ln(7) - \frac{2}{\ln(2)} \ln(7)$ and find the intervals where $f(x) > \frac{4}{\ln(2)}$.

The solution to the inequality is all the $x$ values in those intervals. Note that the exact values of $x$ might be challenging to obtain due to the presence of transcendental numbers (e.g., logarithms), but we can approximate them using numerical methods or graphing tools.

Keep in mind that this solution assumes that $x$ is a real number. If you need a specific numerical solution or a more accurate approximation, you may need to use numerical methods or software to solve the inequality.