Given logx 2=p and logx 7=q , express log7 4x³ in terms of p and q.

(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.