# Logarithms-20

### From Homeworkwiki

**Given log _{ 10 } x = a and log _{ 10 } y = b**

**(i) Write down 10 ^{ a - 1 } in terms of x**

**(ii) Write down 10 ^{ 2b } in terms of y**

**(iii) If log _{ 10 } P = 2a - b ; express P in terms of x and y**

**Solution:**

(i) log _{ 10 } x = a

=> 10 ^{a} = x

Therefore, **10 ^{ a - 1 }** = 10

^{ a }/ 10

^{ 1 }=

**x / 10**

(ii) log _{ 10 } y = b

=> 10 ^{ b } = y

Therefore, **10 ^{ 2b }** = (10

^{ b })

^{ 2 }=

**y**

^{ 2 }(iii) **log _{ 10 } P = 2a - b**

log _{ 10 } P = 2 log _{ 10 } x - log y

=> log _{ 10 } P = log x^{ 2 } - log y

=> log P = log x ^{ 2 } / y

Therefore, **P = x ^{ 2 } / y**