Use the change of base formula to write the following in terms of natural logarithms.

Question

Log₈(3)

 

A. ln(8/3)

B. ln(3/8)

C. ln(3)-3ln(2)

D. 1/3ln(3)/ln(2)

E. 3ln(2)/ln(3)

F. None of the above

Answer 1

Recall that the change of base formula is:

 

log_b c= (log_a c)/log_a b

 

I like writing it as:

 

log_a b · log_b c = log_a c

 

In this form, crossing out b and log_b on the left hand side produces the right hand side.

 

Thus by letting a = e (base of ln) we get:

 

log₈ 3 = (ln 3) / (ln 8) = (ln 3) / (ln 2³) = (ln 3) / (3 ln 2)

 

So it is D.