Definition of Exponential Function to Base aIf 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 ax and is defined by

ax = e (ln a)x

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

These functions obey the usual laws of exponents:

1. a0 = 1

2. axay = ax + y

3. ax = ax - y

ay

4. (ax)y = axy

Logarithmic functions to bases other than e can be defined in much the same way as exponential functions to other bases are defined.

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 logax and is defined as

logax = ln x ln a

Log functions to the base a have properties similar to those of the natural log function:

1. loga 1 = 0

2. loga xy = loga x + loga y

3. loga xn = n loga x

4. loga x = loga x – loga y y

f(x) = ax and g(x) = loga xare inverse functions

Properties of Inverse Functions

1. y = ax if and only if x = loga y

2. alogax = x, for x > 0

3. loga ax = x, for all x

Examples:

Solve for x.

1.3x = 1 81

2. log2 x = -4

x = -4Answer

x = 1/16Answer

Derivatives for Bases Other than e

1. d [ax ] = (ln a)ax

dx

2. d [au ] = (ln a)au du

dx dx

3. d [loga x ] = 1 dx (ln a)x

4. d [loga u ] = 1 du dx (ln a)u dx

Examples:

Find the derivative of each function:

1.y = 2x

2.y = 23x

3.y = log10 cosx

y’ = (ln 2)2xAnswer

y’ = (3ln 2)23x

y’ = -1 tanx ln 10

Answer

Answer

Integration of an Exponential Function to a Base Other than e

∫ ax dx = 1 ax + C ln a

Find: ∫ 2x dx

= 1 2x + C ln 2Answe

r

Review of The Power Rule for Real Exponents:

Let n be any real number and let u be a differentiable function of x.

1. d [xn] = nxn – 1

dx

2. d [un] = nun – 1 du dx dx

Find the derivative of each below:1.y = ee

2.y = ex

3.y = xe

4.y = xx

y’ = 0

y’ = ex

y’ = exe - 1

y’ = xx(1 + ln x)

Answer

Answer

Answer

Answer