Prove that A is a constant A = (log_5_x^2 + log_5_x^3)/(log_4_x^2 + log_4_x^3)

To prove that A is a constant means that to prove that the
result of the ratio does not depend on x.

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

We'll use
the rule of product:

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

log_5_x^2 + log_5_x^3 =
log_5_(x^2*x^3)

log_5_(x^2*x^3) =
log_5_x^(2+3)

log_5_x^2 + log_5_x^3 =
log_5_x^5

We'll use the power rule of
logarithms:

log_5_x^5 = 5*log_5_x
(1)

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

log_4_x^2 + log_4_x^3 =
log_4_x^5

log_4_x^2 + log_4_x^3 = 5*log_4_x
(2)

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

A = 5*log_5_x/5*log_4_x

We'll
simplify:

A = log_5_x/log_4_x

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

log_4_x = (log_5_x)*(log_4_5)

We'll
re-write A:

A =
log_5_x/(log_5_x)*(log_4_5)

We'll
simplify:

A = 1/log_4_5

A
= log_5_4

As we can notice, the result
is a constant and it's not depending on the variable x.