Given log_a2 = x, log_a 3 = y and log_a 5 = z. Find the value in each of the following in terms of x, y and z.

(i) log_a15 (ii) log_a8 (iii) log_a30

(iv) (v) (vi) log_a1.5

(i) log_a15 = log_a(5×3)

i.e. log_a(5×3) = log_a5 + log_a3

(∵ log_a(M×N) = (log_aM) + (log_aN))

= z + y(∵ log_a5 = z,log_a3 = y)

(ii) log_a8 = log_a2³ = 3log_a2 = 3x

(∵ log_a2 = x)

(iii) log_a30 = log_a(5×3×2) = log_a(5) + log_a(3) + log_a(2)

(∵ log_a(M×N) = (log_aM) + (log_aN))

= z + y + x

(∵ log_a5 = z,log_a3 = y,log_a2 = x)

= x + y + z

(iv)

⇒ log_a(3×3×3) - log_a(5×5×5)

⇒ (log_a3 + log_a3 + log_a3) - (log_a5 + log_a5 + log_a5)

⇒ (y + y + y) - (z + z + z) = 3y - 3z = 3(y - z)

(v)

⇒ log_a10 - log_a3

(∵ log_a(M ÷ N) = log_aM - log_aN)

Here log_a10 = log_a(5×2)

(∵ log_a(M×N) = (log_aM) + (log_aN))

= log_a5 + log_a2 = z + x (∵ log_a5 = z,log_a2 = x)

(vi)

⇒

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

= y - x(∵ log_a3 = y,log_a2 = x)