SOLUTION: i really don't know where to begin with this problem at all? given log base a 2= 0.431 and log base a 3= 0.683 find the following .... log base a 6 and log base a 81 and log bas

Question 115571: i really don't know where to begin with this problem at all?
given log base a 2= 0.431 and log base a 3= 0.683 find the following ....
log base a 6 and log base a 81 and log base a 2a?

if you could help me it would be great thank you !
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
You need to know a couple of properties of logarithms.
.
One of the properties is the multiplication property:
.
The first problem is to find . Note that 6 is equal to 2*3. So you can re-write
the problem as But the property of logs described above converts this to:
.

.
Both of these logs are given ... and Inserting
these values into the problem results in:
.

.
So the answer is
.
The next problem uses the exponent property that
.
The problem is to find the value of . To do that you can recognize that
. If you substitute that into the problem, you get:
.

.
Next apply the exponent property to add another step to the problem:
.

.
Finally add the last step by substituting 0.683 for :
.

.
So the answer to this second problem is
.
The final problem is:
.

.
Use the multiplication property to split this into two logarithms as follows:
.

.
Another property is that for any base if you take the logarithm of the base, the answer is 1.
This means . Plus you are given that . Making these
two substitutions results in:
.
.
.
So the answer to this problem is:
.
Hope this helps you to gain some insight into a couple of properties of logarithms. It's
sort of late to be doing math, so be sure to check the math in these problems. Mistakes
have a tendency to creep into late-night work ... but the basic principles involving the
properties of logarithms are correct.
.
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