Question

Решить показательное неравенство:

Answer 1

$4^{2x-1}<7^{1-2x};$
$4^{2x-1}<(\frac{1}{7})^{2x-1}$
$4^{2x-1}*7^{2x-1}<1$
$(4*7)^{2x-1}<1$
$28^{2x-1}<28^0$
$28>11; 2x-1<0;$
$x<\frac{1}{2}$
ответ: x є $(-\infty;0.5)$

Answer 2

Solving the Exponential Inequality

To solve the exponential inequality \(log_3^2x-10log_3x\geq-21\), we can follow these steps:

Step 1: Simplify the inequality by applying the properties of logarithms. In this case, we can use the property \(log_a^n = n \cdot log_a(x)\) to rewrite the inequality as:

\(2log_3x - 10log_3x \geq -21\)

Step 2: Combine like terms:

\(-8log_3x \geq -21\)

Step 3: Divide both sides of the inequality by -8. Remember that when dividing or multiplying both sides of an inequality by a negative number, the direction of the inequality sign must be reversed:

\(log_3x \leq \frac{21}{8}\)

Step 4: Convert the logarithmic inequality into an exponential inequality. Since the base of the logarithm is 3, we can rewrite the inequality as:

\(x \leq 3^{\frac{21}{8}}\)

Step 5: Simplify the exponential expression:

\(x \leq \sqrt3^{21}}\)

Step 6: Evaluate the exponential expression:

\(x \leq 15.34\)

Therefore, the solution to the exponential inequality \(log_3^2x-10log_3x\geq-21\) is \(x \leq 15.34\).