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