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

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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.**