Why do some logarithmic equations have two solutions?

Question

I was trying to find solutions for a high school math problem, but there was one thing I didn't quite understand.

There is a logarithmic identity that says that

$ln\:x^2=2\cdot ln \:x$

However, when solving an equation, the two different forms give different solutions When graphing in Geogebra, or trying to solve with wolframalpha, $ 2 \cdot ln\:x=1$ has only a positive answer to the equation ($\sqrt e$)

But when using $ln\:x^2=1$, I also get the solutions for negative x ($\:\sqrt e, -\sqrt e)$.

If these forms are exactly the same, why do they give different solutions?

score 4 · Accepted Answer · answered Mar 17 '19 at 19:42

4

The correct identity is

$$\log x^2=2\log |x|.$$

answered Mar 17 '19 at 19:42

Thanks! It's weird that it's not mentioned anywhere in the book – Zack King Mar 17 '19 at 19:45
@ZackKing: this is because $x^2=|x|^2$, nothing to do with logarithms. – Mar 17 '19 at 19:47

score 2 · Answer 2 · answered Mar 17 '19 at 19:42

2

$\ln(x^2) = 2\ln(x)$ only holds for $x > 0$.

answered Mar 17 '19 at 19:42

Klaus

10,578

It does really seem like that, but shouldn't the texbook at least mention that? Thank you for the answer though :)) – Zack King Mar 17 '19 at 19:43
I agree, but I guess this is implied as $\ln(x)$ was probably only defined for $x > 0$ anyway. – Klaus Mar 17 '19 at 19:45

Why do some logarithmic equations have two solutions?

2 Answers2