Question

Given log₁₀ x= a, log₁₀ y = b and log₁₀ z =c,

1. write down 10^2a-3 in terms of x.
2. write down 10^3b-1 in terms of y.
3. if log₁₀ P = 2a + b/2 – 3c, express P in terms of x, y and z.

Answer 1

Given:

log₁₀ x= a

• (10)^a = x

log₁₀ y = b

• (10)^b = y

log₁₀ z =c

• (10)^c = z

1. Write down 10^2a-3 in terms of x.

   10^2a-3 = (10)^2a / (10)³

   = (10^a)² / (10 × 10 × 10)

   Substitute the value of (10)^a = x, we get

   = x²/1000 .......(1 mark)

2. Write down 10^3b-1 in terms of y.

   10^3b-1 = (10)^3b / (10)¹

   = (10^b)³ / (10)

   Substitute the value of (10)^b = y, we get

   = y³/10 ........(1 mark)

3. If log₁₀ P = 2a + b/2 – 3c, express P in terms of x, y and z.
   
   we know that,

   (10)^a = x

   (10)^b = y

   (10)^c = z

   By substituting the values

   log₁₀ P = 2a + b/2 – 3c

   = 2 log₁₀ x + ½ log₁₀ y – 3 log₁₀ z

   = log₁₀ x² + log₁₀ y^1/2 – log₁₀ z³

   = log₁₀ (x² + y^1/2) – log₁₀ z³

   = log₁₀ [(x²√y)/z³]

   P = (x²√y)/z³ ........( 2 mark)