Question

Y=log0,3(1-4x) найдите область определения

Answer 1

У=lg0,3(1-4x)
1-4x>0
-4x>-1
x<1/4
x€(- бес ; 1/4)

Answer 2

To find the domain of the function $Y = \log_{0.3}(1-4x)$, we need to consider the conditions for the logarithm to be defined.

The logarithm function $\log_b(x)$ is defined only for positive values of $x$ and the base $b$ must be greater than 0 and not equal to 1.

In this case, we have $b = 0.3$, and the argument of the logarithm is $1-4x$.

So, for the function $Y$ to be defined, the following conditions must be satisfied:

1. $1 - 4x > 0$ (the argument inside the logarithm must be positive)
2. $0.3 > 0$ (the base of the logarithm must be greater than 0)
3. $0.3 \neq 1$ (the base of the logarithm must not be equal to 1)

Let's solve these conditions one by one:

1. $1 - 4x > 0$: To find the interval where this condition is satisfied, solve the inequality for $x$: $1 - 4x > 0$ $4x < 1$ $x < \frac{1}{4}$

2. $0.3 > 0$: This is true as $0.3$ is greater than $0$.

3. $0.3 \neq 1$: This is also true as $0.3$ is not equal to $1$.

Now, we need to combine the results from conditions 1 and 2:

The domain of $Y = \log_{0.3}(1-4x)$ is the set of all $x$ values that satisfy both conditions, which is:

$x < \frac{1}{4}$

Therefore, the domain of the function is $x$ such that $x < \frac{1}{4}$.