Demonstrate that the fraction is a constant [log(x^2) + log(x^3)]/[ln(x^2)+ln(x^3)].

We'll prove that simplifying the fraction, the result does
not depend on the variable x.

We notice that the numerator
is a sum of logarithms that have matching bases.

We'll
apply the product rule:

log a + log b = log
(a*b)

log (x^2) + log (x^3) =
log (x^2*x^3)

log (x^2*x^3)= log
[x^(2+3)]

log (x^2) + log (x^3)
=  log (x^5)

We'll use the power rule of
logarithms:

log (x^5)= 5*log (x)
(1)

We also notice that the denominator is a sum of
logarithms that have matching bases.

[ln (x^2) + ln
(x^3)] = ln (x^5)

[ln (x^2) +
ln (x^3)] = 5*ln x (2)

We'll
substitute both numerator and denominator by (1) and
(2):

5*log (x) /5*ln x

We'll
simplify:

log (x) /ln x

We'll
transform the base of the numerator, namely 5, into the base
4.

ln x  = log (x) * ln
10

We'll re-write the
fraction:

log (x) /ln x = log (x) / log (x) * ln
10

We'll simplify:

1/ln
10

Since the result is not depending on the
variable x, the given fraction is a constant.