Use the definition of logarithm to simplify each expression.
(a) )log_(3b) \(3b\)
((b) )log_(8b) \((8b)^6\)
(c) )log_(10b) \((10b)^(-13)\)
To simplify these expressions using the definition of logarithm, we need to apply the logarithmic properties. Specifically, logarithm with the same base can be multiplied, divided, or exponentiated according to the properties:
(a) Applying the logarithmic property, we have:
log base (a) of (a^b) = b
Therefore, applying this property, we get:
log base (3b) of (3b) = 1
(b) Similarly, using the logarithmic property, we have:
log base (a) of (a^b) = b
Following this property, we can simplify as follows:
log base (8b) of (8b)^6 = 6
(c) Once again, utilizing the logarithmic property, we have:
log base (a) of (a^b) = b
Using this property:
log base (10b) of (10b)^(-13) = -13
Therefore, the simplified expressions are:
(a) 1
(b) 6
(c) -13