Simplifying $\ln$ equations!

Question

If you have an equation such that $\ln$ is on both sides, for example

$\ln x = \ln t$ and you want to solve for $x$, could you say $x = t$?

And if you have another equation such that $\ln$ is on both sides, but one is squared,

$\ln^2 (x) = \ln t$, could the solution if solving for $t$ be $t = \ln x$?

I can understand the first example but I am not understanding the math behind the second one, how does it end up as $t = e^{\ln^2 x}$? (unless the online calculators are wrong) Since $\ln$ is on both sides would you need to use e on the right side?

Note that $ e^{ \ln^2 (x)} = x^{\ln (x)} $... so $\ln^2 (x) = \ln t$ implies that $t=x^{\ln (x)} $ — user577215664, Aug 30 '17 at 23:27

Siong Thye Goh · Answer 1 · 2017-08-30T23:07:21.697

2

For your first question,

$$\ln x = \ln t$$

we can apply $\exp(.)$ on both sides and obtain $x=t$.

Now, for your second question.

$$\ln^2(x) = \ln t$$

We first address what does $\ln^2$ mean. We can refer to the discussion here. Usually it means $$(\ln (x))^2 = \ln t$$

Using the same trick as earlier, we can apply $\exp(.)$ on both sides and obtain

$$t = \exp((\ln(x))^2)$$

Note that $\ln^2 (x) \neq \ln(\ln (x))$, if the question were to be

$$\ln(\ln (x)) = \ln t,$$

upon applying $\exp(.)$ on both sides, we have

$$t = \ln (x).$$

edited Aug 30 '17 at 23:07

answered Aug 30 '17 at 23:02

Siong Thye Goh

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According to the OP, the online calculator gives $e^{\ln^2 x}$, exactly as you have. $t = \ln x$ is speculation on the OP's part. – Erick Wong Aug 30 '17 at 23:04
ah, i see, i misread the situation. Thanks. – Siong Thye Goh Aug 30 '17 at 23:05

Simplifying $\ln$ equations!

1 Answers1