Answer
a. The equation has at least one real root.
b. The root is $1.35$.
Work Step by Step
$\ln x = 3 -2x$ Equal this function to $f(x)$.
$f(x) = \ln x +2x - 3$
From the definition of limits involving $\ln$ we know that:
$\lim\limits_{x \to \infty} \ln x = \infty$ and $\lim\limits_{x \to 0} ln x = -\infty$.
So we note that in this problem:
$\lim\limits_{x \to 0^{+}} \ln x +2x - 3 = -\infty$
$\lim\limits_{x \to \infty} \ln x +2x - 3 = \infty$
We know from the intermediate value theorem that $f(x)$ takes all values between $-\infty$ and $+\infty$. So there is at least one root for $f(x)$.
b. To find the interval make one graph using $(\ln x)$ and $(3-2x)$ From the graph we can see that the root lies in 1.35
