Help,can anyone show me how to do this? find the value using only the properties of logarithms, no calculator

Question

find thevalue using the properties of logarithms no calculators, how?

2log subscript6 3+ log subscript6 4
the 2, 3, 4, are not subscrpited or superscripted. only both 6 are subscripted.
thanks, i am studying for a test.

Answer 1

2log (base 6) 3+ log (base 6) 4
= log (base 6) 3^2 + log (base 6) 4
= log (base 6) 3^2 * 4
= log(base 6) 36
= log(base 6) 6^2
= 2 log(base 6) 6
= 2 * 1
= 2

Answer 2

Do you remember you logarithm laws?

(A) log (ab) = log(a) + log(b)
(B) log(a^k) = klog(a)

which apply for any base (you are using base 6 in your question).

Using (B) on 2log 6(3) gives log 6(3^2) = log 6 (9), and adding to log 6(4) using (A) gives log 6(9*4) = log 6(36).

Finally, remember that logarithms and powers "undo" the effects of each other (are inverse functions). So, for example, when we write 10^2 = 100, this is the same as saying log 10(100) = 2. It should be clear then that using base 6
log 6(36) = log 6(6^2)
= 2.

Answer 3

First, 2 log_6 3 = log_6 3² = log_6 9.
Then log_6 9 + log_ 6 4 = log_6(9*4) = log_6 36.
But log_6 36 = 2, since 6² = 36.
The answer is 2.