Logarithmic equations: Solving for the Variable

Question

The equation is $(3/2)\ln x= -2$. I am not sure how to work this one. If anyone could show all the steps that would be a great help. I tried working it out and got down to $x^{3/2}=e^{-2}$ that is probably wrong but if by some chance it is right what do you do after that?

Answer 1

That $x^{3/2}=e^{-2}$ is correct. If you know $x^{3/2}=\text{something}$, then you can say $$ \Big(x^{3/2}\Big)^{2/3} = \text{something}^{2/3} $$ and then $$ x^{(3/2)\cdot(2/3)} = \text{something}^{2/3}. $$

Then remember how to simplify $\dfrac 3 2 \cdot \dfrac 2 3$.

Then you need to simplify $\displaystyle\Big(e^{-2}\Big)^{2/3}$. That is similar to something done above.

Answer 2

$$ \frac{3}{2} \ln x = -2 \iff \ln x = \frac{ -4}{3} \iff e^{-\frac{4}{3} }= x$$

Answer 3

$$\dfrac{3}{2}\ln x=-2\\ \implies \ln x=\dfrac{-4}{3}\\ \implies \exp(\ln x)=\boxed{x=\exp\left(\dfrac{-4}{3}\right)}$$