Use common logarithms or natural logarithms

Question

Use common logarithms or natural logarithms to evaluate to four decimal places. 

 

log_π15 

 

 

 

 

 

 

 

 

 

(use the change-of-base property) 

Answer 1

log_π15 = log 15 / log π

log_π15 = 1.176 / 0.497

log_π15 = 2.366

Answer 2

Cathryn,

 

The Log _(base) N  = k  means that you have to raise the base to the k^th power to get the number N.

 

If base = 10 , then    (10)^k = 15. Solve this via a log₁₀ table lookup or on calculator.  So k = 1.1761

 

If base = e = 2.718282, then  (2.718282)^k = 15  so k = 2.7081 using natural log table or calculator function.

 

To use the change of log base property, say we want to go from base a = 10 to base b = e:

Then  Log _b N = (Log _b a) (Log _a N)

         Log _e 15 = (Log _e 10) (Log ₁₀ N)

                      = (2.30259) (1.17609) = 2.7081 as above

Similarly, you can go the other way just by switching your a and b values.

 

Thus, the same number (15) is represented differently depending on the base used.  In any problem if you have to use logarithms, pick a base and stick with it to avoid errors and confusion.