Hello, I've trouble solving "graph log2x<0" how do i go about doing this? I know that log2x=y is equivalent to 2^y=x. However, the inequality sign and the <0 makes it complicated. The given answer is 1>x>0.
If , then . If you increase , gets "closer" to zero. You should know that there is no value for that makes the equation true, so is not defined. Thus, . Instead of increasing , let's decrease it. As gets closer to zero, gets closer to 1, but is always less than one. So, that tells you .
You graph this inequality the exact same way you graph ANY inequality.
First, you graph the related equality. In this case that seems to be $y = log_2(x).$ Is that function defined for non-positive values of x?
Second, you identify on what side of the point or points of equality, the inequality is true.
So at what value of x does $log_2(x) = 0.$
To the left of that point what is the sign of the function? To the right of that point what is the sign of the function?
Well the domain of $\displaystyle \begin{align*} \log_2{(x)} \end{align*}$ is $\displaystyle \begin{align*} x > 0 \end{align*}$. Solving the inequality for x gives
$\displaystyle \begin{align*} \log_2{(x)} &< 0 \\ x &< 2^0 \\ x &< 1 \end{align*}$
Thus, when combined with the domain of this function, the solution must be $\displaystyle \begin{align*} 0 < x < 1 \end{align*}$.
This has only the single variable x, rather than, say x and y. Its graph will be a portion of a number line, not a two dimensional x, y graph. In general, the regions where f(x)< a and f(x)> a are separated by points where f(x)= a. Here, mark the single point x such that . Then, perhaps, choose one value of x less than that and one value of x larger to determine which side satisfies and which satisfies .