7.2* Natural Logarithmic Function In this section, we will learn about: The natural logarithmic function and its derivatives. INVERSE FUNCTIONS

If x and y are positive numbers and r is a rational number, then: LAWS OF LOGARITHMS Laws 3

Expand the expression  Using Laws 1, 2, and 3, we get: LAWS OF LOGARITHMS

Express l n a + ½ l n b as a single logarithm.  Using Laws 3 and 1 of logarithms, we have: LAWS OF LOGARITHMS Example 3

Differentiate y = l n(x 3 + 1).  To use the Chain Rule, we let u = x  Then, y = l n u.  Thus, NATURAL LOG. FUNCTION Example 5

In general, if we combine Formula 2 with the Chain Rule as in Example 5, we get: NATURAL LOG. FUNCTION Formula 6

Find:  Using Formula 6, we have: NATURAL LOG. FUNCTION Example 6

Differentiate:  This time, the logarithm is the inner function.  So, the Chain Rule gives: NATURAL LOG. FUNCTION Example 7

Find: NATURAL LOG. FUNCTION E. g. 8—Solution 1

If we first simplify the given function using the laws of logarithms, the differentiation becomes easier:  This answer can be left as written.  However, if we used a common denominator, we would see it gives the same answer as in Solution 1. NATURAL LOG. FUNCTION E. g. 8—Solution 2

Differentiate:  We take logarithms of both sides and use the Laws of Logarithms to simplify: LOGARITHMIC DIFFERENTIATION Example 14

 Differentiating implicitly with respect to x gives:  Solving for dy/dx, we get: LOGARITHMIC DIFFERENTIATION Example 14

 Since we have an explicit expression for y, we can substitute and write: LOGARITHMIC DIFFERENTIATION Example 14

If we hadn’t used logarithmic differentiation in Example 14, we would have had to use both the Quotient Rule and the Product Rule.  The resulting calculation would have been horrendous. LOGARITHMIC DIFFERENTIATION Note

1.Take natural logarithms of both sides of an equation y = f(x) and use the Laws of Logarithms to simplify. 2.Differentiate implicitly with respect to x. 3.Solve the resulting equation for y’. STEPS IN LOGARITHMIC DIFFERENTIATION