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7.1 The Logarithm Defined as an Integral, Part II The General Exponential Function ax
Last time, we defined ln x =
∫ (^) x
1
t dt
and ex^ to be the inverse of ln x. How to define ax^ and logxa for a positive number a? Since a = eln^ a,
we can think of ax^ = (eln^ a)x^ = ex^ ln^ a.
When a = e, the definition gives ax^ = ex^ ln^ a^ = ex^ ln^ e^ = ex·^1 = ex. Derivative of ax
(ax)′^ = (ex^ ln^ a)′^ = ex^ ln^ a^ · (x ln a)′^ = ax^ ln a
Thus, (ax)′^ = ax^ ln a By applying the Chain Rule, we get a more general form:
If a > 0 and u is a differentiable function of x, then au^ is a differentiable function of x and d dx au^ = au^ ln a du dx The integral equivalent of this last result is (^) ∫
audu = au ln a
+ C
Logarithms with Base a
For any positive number a 6 = 1, loga x is the inverse function of ax.
Since logxa and ax^ are inverses of one another, we have aloga^ x^ = x, (x > 0)
Example: Find d dx log 10 (5x^2 − 2 x + 100)
Example: Find the derivative of log 5 ( √xx+5− 2 ).
Example: Evaluate (^) ∫ √ log 10 x x dx
Example: Evaluate (^) ∫ π 4 0
(^1
)tan^ t^ sec^2 tdt
Example: Evaluate (^) ∫ √ ln π 0
2 xex 2 cos(ex 2 )dx
Example: Evaluate (^) ∫ 10
101
log 10 (10x) x dx
