MAT.126 5.5

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MAT.126 5.5 Bases Other Than e and Applications

Differentiation and Integration

A comparison of 4 rules for differentiation

Constant Rule

d/dx[e^e] = 0

Power Rule

d/dx[x^e] = e x^{e-1}

Exponential Rule

d/dx[e^x] = e^x

Logarithmic differentiation

d/dx[x^x] = x^x (1 + ln x)

Integration

When confronted with an integral of the form

S a^x dx

there are two choices.

One is to convert the exponential expression with base a to an equivalent exponential expression with base e. That is, consider

S e^{(ln a)x} dx

remembering that ln a is a constant.

The second option is to use the following integration formula,

S a^x dx = (1/ln a) a^x + C

Derivatives for Bases Other Than e

Let a be a positive real number (a<>1) and let u be a differentiable function of x.

d

dx

d du

dx dx

d 1

dx (ln a) x

d 1 du

dx (ln a) u dx

Applications of Exponential Functions

Logisitic Growth

Continuously Compounded Interest

Compound Interest

Bases Other than e

Common Logarithmic Function

The logarithm with base 10.

Inverse Function Properties

Logarithmic Properties

Definition of Logarithmic Function to Base a

If a is a positive real number (a<>1) and x is any positive real number, then the logarithmic function to the base a is denoted by log_a x and is defined as

log_a x = (1/ln a) ln x

Laws of Exponents

Definition of Exponential Function to Base a

If a is a positive real number (a<>1) and x is any real number, then the exponential function to the base a is denoted by a^x and is defined by

a^x = e^{(ln a)x}

If a=1, then y=1^x=1 is a constant function.