Defining Logarithms, Study notes of Calculus

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Goals

We will provide a formal, calculus based definition for the natural logarithm, use this definition to re-prove logarithmic properties

The Original Definition

log b ( a ) ≡ x iff bx^ = a.

Calculate the following. 1 log 2 (8) = 2 log 3 (9) = 3 log 5 (10) =

Properties

ln x ≡

∫ (^) x 1

t dt. Use the definition and derivative and integral knowledge to calculate the following. 1 ln 1 = 2 d^ dx ln^ x =

Shape

ln x =

∫ (^) x 1

t dt.

What is ln 1? Remember that an integral adds. Which is bigger ln 1 or ln 2? Which is bigger ln 1 or ln a where a > 1? Which is bigger ln 1 or ln a where a < 1? Which is bigger ln a or ln b where b > a? What does this say about the graph of ln x?

Product Properties

g ( x ) = ln( ax ) , f ( x ) = ln x.

Calculate f ′( x ) and g ′( x ). Compare f ′( x ) and g ′( x ). What does this imply about these functions and their graphs? Evaluate your conclusion at x = 1 and substitute this result into the conclusion.

More Anti-Derivatives

1 Calculate the first derivative of f ( x ) = − ln(cos x ). 2 Calculate the first derivative of f ( x ) = ln(sec x + tan x ). 3

∫ tan x dx 4 ∫^ sec x dx 5

∫ csc x dx