Question

Show that:
$\log \frac{75}{16} - 2\log \frac{5}{9} + \log \frac{32}{243} = \log 2$

Answer 1

Answer:

Step-by-step explanation:

Hi,  

We will be using the following properties of  

logarithm:

Additive Property : logₐx + logₐy = logₐ(xy) ,

Subtraction Property : logₐx - logₐy = logₐ(x/y)and

Exponent Property : nlogₐx = logₐxⁿ

Consider L.H.S

= log (75/16) - 2 log (5/9) + log (32/243)

Using Exponent Property, we can write

2 log (5/9) as log (5/9)² = log 25/81

Now, L.H.S

= log (75/16) - log 25/81 + log (32/243)

= [log (75/16) - log 25/81] + log (32/243)

Using Subtraction Property

= log [(75/16)/(25/81)] + log (32/243)

= log ( 75*81/16*25) + log (32/243)

= log (243/16) + log (32/243)

Using Additive Property

= log (243/16)*(32/243)

= log 2

= R.H.S

Hope, it helps !