Manipulation properties of the logarithm, Algebra

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For these properties we will suppose that x > 0 and y > 0

logb ( xy ) = logb x + logb  y

logb (    x/y) = logb  x - logb  y

logb (xr ) = r log  x

If logb x = logb  y then x = y .

We won't be doing anything along the final property in this section; this is here only for the sake of completeness. We will be looking at this property thoroughly in a couple of sections.

The first two properties illustrated here can be a little confusing primary since on one side we've got a product or quotient within the logarithm & on the other side we've got a sum or disparity of two logarithms.  We will only have to be careful with these properties & ensure to them correctly.

Also, note that there are no rules on how to break up the logarithm of the sum or difference of two terms.  To be apparent about this let's note the following,

logb ( x + y ) ≠ logb  x + logb  y

logb ( x - y ) ≠ logb  x - logb  y

Be careful along with these and do not attempt to use these as simply they aren't true.

Note that every property given to this point is valid for the common and natural both logarithms.  We only didn't write them out explicitly by using the notation for these 2 logarithms, the properties do hold for them nonetheless

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