Logarithm Help

Question

2log base 4 9 - log base 2 3

Answer 1

2 log₄ 9 - log₂ 3

 

Use the Change of Base Formula on 2 log₄ 9 

 

log_b x = log_d (x)/log_d (b), so:

 

log₄ 9 = log₂ (9)/log₂ (4)

 

y = log_a x means a^y = x, so:

 

y = log₂ 4 

 

2^y = 4

 

y = 2

Therefore, log₂ 4 = 2

 

log₂ 9 = log₂ (3²) = 2 log₂ 3

 

We now have log₄ 9 = [log₂ (9)]/log₂ (4) = [2 log₂ (3)]/2 = log₂ 3

 

So 2 log₄ 9 - log₂ 3 = 2 log₂ 3 - log₂ 3 = log₂ 3

 

 

Answer 2

Since 4 is a power of 2

(4 = 2² )

we can change the first base 4 to base 2 by taking square root of 9:

 

2log₄ 9 - log₂ 3  

= 2log₂ 3 - log₂ 3   now both terms have the same log to factor out

= (2-1) log₂ 3

= log₂ 3   {solution}