Question

Show that:
$\log_{v} \sqrt[4]{u^{3}} \cdot \log_{w} v^{5} \cdot \log_{u} \sqrt[5]{w^{4}} = 3$

Answer 1

Answer:

Step-by-step explanation:

Hi,  

Here we will be using the following properties of  

logarithm:  

logₐb = log b/log a

nlog a = log aⁿ

Consider L.H.S

logv(⁴√u³). logw(v⁵).logu(⁵√w⁴)

logv(⁴√u³) = log (⁴√u³)/log v

log (⁴√u³) = 3/4 log u

So, logv(⁴√u³) = 3/4 log u/log v

logw(v⁵) = log v⁵/log w

log v⁵ = 5 log v

So,  logw(v⁵) = 5 log v/log w

logu(⁵√w⁴) = log (⁵√w⁴)/log u

log (⁵√w⁴) = 4/5 log w

So, logu(⁵√w⁴) = 4/5 log w/log u

Substituting the above in L.H.S, we get

= ( 3/4 log u/log v)*( 5 log v/log w)*(4/5 log w/log u)

= 3/4 * 5 * 4/5

= 3

Hope, it helps !