Question

2log1/5(x-2)+3log5(x-2)<1

Answer 1

$2 log_{ \frac{1}{5} } (x-2)+3log_{5}(x-2)\ \textless \ 1 \\ \\ ODZ:x\ \textgreater \ 2 \\ \\ -2log_{5}(x-2)+3log_{5}(x-2)\ \textless \ 1 \\ \\ log_{5}(x-2)\ \textless \ 1 \\ \\ log_{5}(x-2)\ \textless \ log_{5}5 \\ \\ 5\ \textgreater \ 1 \\ \\ x-2\ \textless \ 5;x\ \textless \ 7$

с учетом ОДЗ, получаем ответ :x∈(2;7)

Answer 2

To solve the inequality $2\log_{\frac{1}{5}}(x-2) + 3\log_5(x-2) < 1$, we'll use logarithm properties to simplify it and then find the solution. Let's break it down step by step:

Step 1: Combine the logarithms using logarithm rules Recall the logarithm rule: $a\log_b(c) = \log_b(c^a)$. Apply this rule to the given inequality:

$2\log_{\frac{1}{5}}(x-2) + 3\log_5(x-2) = \log_{\frac{1}{5}}((x-2)^2) + \log_5((x-2)^3)$.

Step 2: Combine the logarithms into a single logarithm Another logarithm rule states that $\log_b(c) + \log_b(d) = \log_b(cd)$. Apply this rule to the logarithms in the inequality:

$\log_{\frac{1}{5}}((x-2)^2) + \log_5((x-2)^3) = \log_{\frac{1}{5}}((x-2)^2 \cdot 5^3) = \log_{\frac{1}{5}}(125(x-2)^2)$.

Now the inequality becomes:

$\log_{\frac{1}{5}}(125(x-2)^2) < 1$.

Step 3: Convert the logarithmic inequality into an exponential inequality The logarithmic inequality can be converted to an exponential inequality as follows:

$\log_b(y) < c$ is equivalent to $y < b^c$.

Apply this to the inequality:

$125(x-2)^2 < \left(\frac{1}{5}\right)^1$.

Step 4: Simplify the exponential inequality $125(x-2)^2 < \frac{1}{5}$.

Step 5: Solve for x Let's solve the inequality:

$125(x-2)^2 < \frac{1}{5}$.

First, divide both sides by 125:

$(x-2)^2 < \frac{1}{5 \cdot 125}$.

$(x-2)^2 < \frac{1}{625}$.

Now, take the square root of both sides. Remember to consider both the positive and negative square roots:

$x-2 < \pm \sqrt{\frac{1}{625}}$.

$x-2 < \pm \frac{1}{25}$.

Now, we have two separate inequalities to consider:

1. $x-2 < \frac{1}{25}$.

Add 2 to both sides:

$x < \frac{1}{25} + 2$.

$x < \frac{51}{25}$.

2. $x-2 > -\frac{1}{25}$.

Add 2 to both sides:

$x > -\frac{1}{25} + 2$