Question

Simplify 7 log3 k + 6 log3 m − 9 log3 n.

4 log3 km over n

4 log3 (k + m − n)

log3 k to the seventh power m to the sixth power over n to the ninth power

log3 42 km over 9 n

Answer 1

These are fun! Do you still need help with this?

Answer 2

yes i do

Answer 3

Ok, since these all have the same base of 3 we can easily do this.

Answer 4

ok

Answer 5

When logs are added up there, that means that once upon a time, they were multiplied. When two logs are subtracted, that means that once upon a time, before they were expanded, they were divided. Let's start with the 7 the 6, and the 9 in front of all the terms.

Answer 6

how do we start

Answer 7

Those were exponents before they were expanded, so let's put them back that way. The rule for that is \[\log _{a}x ^{p}=p * \log _{a}x\]

Answer 8

Let's use that rule first:

Answer 9

\[\log _{3}(k)^{7}+\log _{3}(m)^{6}-\log _{3}(n)^{9}\]

Answer 10

See how that rule applied to what we had and now those numbers are written as exponents?

Answer 11

yes

Answer 12

k, now we will deal with the addition part. The rule for that is this:

Answer 13

\[\log _{a}x + \log _{a}y=\log _{a}(x * y)\]

Answer 14

That changes what we had as an added set of terms to:

Answer 15

\[\log _{3}(k ^{7} * m ^{6})-\log _{3}(n)^{9}\]

Answer 16

Can you follow that part using the multiplication property?

Answer 17

i cant

Answer 18

im bad at math

Answer 19

Ok, the log base is the same, so as long as the log base is the same, when you are adding two logs with the same bases, it's the same as multiplying them. That's just the way it is for logs; it doesn't have to make sense. That's just the rule.

Answer 20

Don't try to understand the WHY of these; they just ARE. "log base a of x + log base a of y= log base a of x * y." That's just the way it is.

Answer 21

would it be 4log3