I'm not understanding this natural log simplification, could someone explain it to me?

Question

On #1 the final answer is 1/2ln|4| and they simplify it to ln(2)

But I don't understand why on #2 they didn't simplify 1/2ln|10| to ln(5)

Am I missing some sort of ln rule?

Answer 1

In the first example, the simplification is $\frac{1}{2} \ln(4) = \ln \left( 4^{1/2} \right) = \ln(2)$. Note that the $1/2$ in front becomes an exponent inside the logarithm, not a denominator. (It's a coincidence that $4^{1/2}$ and $4/2$ are both equal to $2$.)

In the second example, you could, if you want, rewrite the answer as $\ln 10^{1/2}$ or $\ln \sqrt{10}$. Whether or not these are "simpler" is largely a matter of taste and convention.

Answer 2

You have $$a \log(x)= \log(x^a)$$ in your case $\sqrt{4} = 2$. In the second question we would end up with $\sqrt{10}$ which cannot really be simplyfied.

Answer 3

In general, $\frac 12\ln(A) = \ln(A^{1/2}) = \ln(\sqrt{A})$. This simplifies when $A = 4$, but not when $A = 10$.