Answer: (2*p + 3)/q
Step-by-step explanation:
First, let's remember the relationships:
Logₙ(a) = Ln(a)/Ln(n)
Ln(A*B) = Ln(A) + Ln(B)
Ln(a^n) = n*Ln(a)
Now, we know that:
Logₓ(2) = p
Logₓ(7) = q
We want to express:
Log₇(4*x^3) in terms of p and q.
First, we can rewrite the first two relations as:
Ln(2)/Ln(x) = p
Ln(7)/ln(x) = q
then we have:
Ln(2) = p*Ln(x)
Ln(7) = q*Ln(x)
Ok:
Now let's play with our equation:
Log₇(4*x^3)
First, this is equal to:
Ln(4*x^3)/Ln(7)
We now can rewrite this as:
(Ln(4) + Ln(x^3))/Ln(7)
= (Ln(2^2) + Ln(x^3))/Ln(7)
= (2*Ln(2) + 3*Ln(x))/Ln(7)
Now we can replace Ln(2) by p*Ln(x) and Ln(7) by q*Ln(x)
(2*p*Ln(x) + 3*Ln(x))/(q*Ln(x)) = (2*p + 3)/q
This is the expression we wanted.
