Question

If x = 1 + log 2 - log 5, y = 2 log3 and z = log a - log 5; find the value of a if x + y = 2z.

Answer 1

Given that 
x = 1 + log 2 - log 5,
y = 2 log 3 and
z = log a - log 5
Consider
x = 1 + log 2 - log 5
= log 10 + log 2 - log 5
= log( 10 x 2 ) - log 5
= log 20 - log 5
= log ${\displaystyle \frac{20}{5}}$
= log 4                   ....(1)
We have
y = 2 log3
= log 3²
= log 9                  ....(2)
Also we have
z = log a - log 5
= log${\displaystyle \frac{a}{5}}$             ....(3)
Given that x + y = 2z
∴ Subsitute the values of x, y, and z.
from (1), (2) and (3), We have
⇒ log 4 + log 9 = 2 log ${\displaystyle \frac{a}{5}}$ 

⇒ log 4 + log 9 = log${\displaystyle {\left(\frac{a}{5}\right)}^2}$

⇒ log 4 + log 9 = log${\displaystyle \left(\frac{a^2}{25}\right)}$

⇒ ${\displaystyle \log (4\times \log 9)=\log \left(\frac{a^2}{25}\right)}$

⇒ ${\displaystyle \log 36=\log \left(\frac{a^2}{25}\right)}$

⇒ ${\displaystyle \frac{a^2}{25}=36}$
⇒ a² = 36 x 25
⇒ a²  = 900
⇒ a = 30.