1. Section 3.6 Algebra Review
(1) Definition of the logarithm: loga x = y iff
(2) Change of Base Formula: loga u =
(3) Exponent-Log Formula: uw =
(4) Product-Sum Formula: loga (uv) =
(5) Quotient-Difference Formula: loga (u/v) =
(6) Exponent-Product Formula: loga uw =
2. Old Calculus Formulas
(1)
d x
(e ) =
dx
(2)
d x
(a ) =
dx
(3)
d f (x)
(e ) =
dx
(4)
d f (x)
(a ) =
dx
3. New Calculus Formulas
(1)
d
(ln x) =
dx
d
Remark 3.1. The domain of f (x) = ln x is x > 0, so the domain of
(ln x)
dx
is
(2)
d
(loga x) =
dx
(3)
d
(ln f (x)) =
dx
(4)
d
(loga f (x)) =
dx
1
Section 3.6 Logarithms
2
4. Examples
Example 4.1. Find the derivative of y = e2x ln(x3 + e2x )
(log3 x)4
Example 4.2. Find the derivative of f (x) =
x2
Example 4.3. Find the point(s) on the graph of y = x2 ln x at which the tangent line
is horizontal.
Section 3.6 Logarithms
3
(2x + 5)4
Example 4.4. Use the properties of logarithms to find the derivative of y = ln 2 √
x 3x + 1
Example 4.5. Use logarithmic differentiation to find the derivative of y =
(2x + 5)4
√
x2 3x + 1
Section 3.6 Logarithms
Example 4.6. Use logarithmic differentiation to find the derivative of y = x(x
4
2 +1)
Example 4.7. Use logarithmic differentiation to find the derivative of y = (sin x)x cos x