SOLUTION: Can you please try to show your steps on how you got the following answers? Given log[a](5)=2.3 and log[a](3)=1.6, fill in the table below with the appropriate values. x

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Question 594466: Can you please try to show your steps on how you got the following answers?
Given log[a](5)=2.3 and log[a](3)=1.6, fill in the table below with the appropriate values.
x 15 ; 9 ; 5/3 ; 5a ; (3)/(a^2)
log[a](x)
Answer by jsmallt9(3759) (Show Source):
You can put this solution on YOUR website!
Given:

Plus,
(This is always true, no matter what "a" is, so it does not need to be told this.)
you have been asked to find various other base a logarithms. The "trick" to these is to use algebra and/or properties of logarithms to rewrite the desired logs in terms of the logs you already know.

So for

we want to rewrite the 15 in terms of 5's, 3's and/or a's. I hope that you can see that 15 is 5*3. Replacing the 15 with 5*3 we get:

Now we use a property of logarithms for logs of a product, , we can separate the 5 and 3:

Now that we have the log of 15 expressed in terms of logs of 5 and 3. We can now use the given values:
2.3 + 1.6
which simplifies to 3.9. So

For 9 we could use either 3*3 or . I'll use the later one so you can see another property in use:

Using a property for logs of a power, we can separate the exponent from the 3:

Replacing the log with its given value we get:
2 * 1.6
which simplifies to
3.2
So

5/3 is already expressed in terms of 5's and 3's. Using a property for logs of quotients, we get:

Replacing the logs with their given values we get:
2.3 - 1.6
which simplifies to
0.7
So

First we'll use the property for quotients:

and then the property for powers (on the second log):

Now we can replace the logs with their known values.
1.6 - 2*1
which simplifies
1.6 - 2
-0.4
So