Question

If a = 1 + log 2 – log 5, b = 2 log 3 and c = log m – log 5, find the value of m if a + b = 2c.

Answer 1

We are given:

• a = 1 + log 2 – log 5
• b = 2 log 3
• c = log m – log 5
• a + b = 2c

Step 1: Combine expressions for a, b and c

a: `a = 1 + log 2 - log 5 = log 10 + log 2 - log 5 = log((10 * 2)/5) = log 4`

b: `b = 2 log 3 = log 3^2 = log 9`

c: `c = log m - log 5 = log (m/5)`

Step 2: Use the equation a + b = 2c

`log 4 + log 9 = 2 log (m/5)`

Left side:

`log(4 * 9) = log 36`

So:

`log 36 = 2 log (m/5)`

⇒ `log 36 = log(m/5)^2 = log (m^2/25)`

Step 3: Equate arguments

`36 = m^2/25`

⇒ m² = 900

⇒ m = ±30

Step 4: Check validity

• log m is only defined for m > 0, so m = –30 is invalid

m = 30