1. 5.5 Bases Other Than e and Applications

2. ax = e (ln a)x 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 ax and is defined by
ax = e (ln a)x
If a = 1 , then y = 1x = 1 is a constant function

3. These functions obey the usual laws of exponents:
2. axay = ax + y
3. ax = ax - y ay
4. (ax)y = axy

4. logax = ln x ln a Definition of Logarithmic Function to Base a
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

5. loga 1 = 0 loga xy = loga x + loga y loga xn = n loga x
Log functions to the base a have properties similar to those of the natural log function:
loga 1 = 0
loga xy = loga x + loga y
loga xn = n loga x
loga x = loga x – loga y y

6. f(x) = ax and g(x) = loga x
are inverse functions
Properties of Inverse Functions
y = ax if and only if x = loga y
alogax = x, for x > 0
loga ax = x, for all x

7. Examples: Solve for x. 3x = 1 81 2. log2 x = -4 Answer x = -4 x = 1/16

8. Derivatives for Bases Other than e
d [ax ] = (ln a)ax dx
2. d [au ] = (ln a)au du
dx dx
3. d [loga x ] =
dx (ln a)x
4. d [loga u ] = du
dx (ln a)u dx

9. Answer Examples: Find the derivative of each function: y = 2x y = 23x
y = log10 cosx
Answer
y’ = (ln 2)2x
y’ = (3ln 2)23x
Answer
y’ = tanx
ln 10
Answer

10. Integration of an Exponential Function to a Base Other than e
∫ ax dx = ax + C
ln a
Find: ∫ 2x dx
= x + C
ln 2
Answer

11. Review of The Power Rule for Real Exponents:
Let n be any real number and let u be a differentiable function of x.
d [xn] = nxn – 1 dx
2. d [un] = nun – 1 du
dx dx

12. y = ee y = ex y = xe y = xx Find the derivative of each below: Answer
y’ = exe - 1
Answer
y’ = xx(1 + ln x)