properties1 the domain of the logarithm

Question

Properties

1.   The domain of the logarithm function is (0, ∞ ) .  In other terms, we can just plug positive numbers into a logarithm! We can't plug in zero or a negative number.

2.    log_b b = 1

3.    log_b 1 = 0

4.    log_b b^x= x

5.    b^log_b ^x  = x

The final two properties will be especially useful into the next section.  Notice that these last two properties tell us that,

f( x ) = b ^x              and g( x ) = log_b x

are inverses of each of other.

Following are some more properties which are useful in the manipulation of logarithms.

More Properties

6.   log_bxy= log_b  x + log_b  y

7.   log_b (x/y)=       log_b  x - log_b  y

8.   log_b( x^r) = r log_b  x

Note as well that there is no equivalent property to the first two for sums & differences. In other terms,

log_b( x + y ) ≠ logb  x + logb  y

logb(x - y ) ≠ logb  x - logb  y