How do I expand a log with a square root?

Question

How would I expand log₅ (7√x)?

Answer 1

log_5 (7√x), where subscript 5 for the base is shown as _5.

 

Use Product Rule of Logarithms:

 

log_5 (7) + log_5 (√x)

 

Replace √x with x^(1/2) and use Power Rule of Logarithms:

 

log_5 (7) + (1/2)*log_5 (x)

 

That's the full expansion.

Answer 2

Here we have it 5 as base:

 

log _{5 (} 7 √x ) = log₅ 7 + 1/2 log ₅ X        

                     = 1.2 + 1/2 log₅ X                                     5 ^m = 10 ^{log 7}

                                                                                   m log 5 = 0.85 log10 

                                                                                    m = 0.85/ 0.7 = 1.2

 

                                                                                   log₅ 7 = log 7/ log 5= 1.209

 

   Here it is just log 5 *√X 

  

log 5 ( 7 √X )   = log 5 + log 7 + 1/2 log X

                      = log 35 + 1/2 log X   

                       = 1.5 + 1/2 log X