Download Change of Base xf = == 2) 2(5)(

Name
December 18/19, 2014
Math 4 notes and problems
section 3.4 continued page 1
Change of Base
Objective: Develop and use change-of-base procedures for exponents and logarithms.
Using inverses
Reminder: Functions f(x) = bx and g(x) = logb(x) are inverses of each other, which means that
f(g(x)) = x and g(f(x)) = x.
1. All but one of the following expressions can be evaluated instantly without a calculator.
Write the values. Cross out the one that can’t be found instantly.
log5(54) =
log7(7k) =
loga(a8) =
2 log2 (6) =
b logb ( 9 ) =
b logb ( c ) =
log(10–4) =
log2(105) =
e ln(3) =
2. Fill in each box with an expression that makes the equation true.
log 4 (
5=2
)=6
log 3 (
10
) = −5
=e
ln(
b
Exponential change-of-base
Example: Change f(x) = 5x into base 2.
Solution:
f ( x) = 5 x = (2 log2 5 ) x = 2 (log2 5)⋅x
3. Change each function into a function involving a specified base.
a. Change f(x) = 3x into base 10.
b. Change f(x) = 100 · 2x into base e.
c. Change f(x) = bx into base a.
) =1
=a
Name
December 18/19, 2014
Math 4 notes and problems
section 3.4 continued page 2
A new approach to solving equations for exponents
Previously you’ve solved equations such as 5x = 7 by converting into logarithmic form,
x = log5(7), then evaluating on the calculator. However, here’s another way that you can solve
such equations, in which the key step is taking the log of both sides of the equation.
5x = 7
Take log of both sides:
log(5x) = log(7)
Use a log property:
x log(5) = log(7)
Divide:
x =
log(7)
log(5)
In the above, the logs used were base 10 logs, but logs of any base could be used. Here’s the
same procedure using base 3 logs and using base e logs.
5x = 7
5x = 7
log3(5x) = log3(7)
ln(5x) = ln(7)
x log3(5) = log3(7)
x ln(5) = ln(7)
x =
log 3 (7)
log 3 (5)
x=
ln(7)
ln(5)
So, the equation’s solution is the quotient of two logs from whatever base you choose.
4. Solve each equation, starting with the first step specified.
a. 2x = 17
starting with log( ) of both sides
b. 7x = 9
starting with log4( ) of both sides
c. 123x = 456
starting with ln( ) of both sides
d. 8x = 16
starting with any log you choose
Name
December 18/19, 2014
Math 4 notes and problems
section 3.4 continued page 3
Change-of-base formula
The change-of-base formula for logarithms expresses the idea that the solution to the equation
bx = c, notated as x = logb(c), can be expressed as a quotient of logs in any chosen base d.
Change-of-Base Formula:
log b (c) =
log d (c)
log d (b)
5. Prove the Change-of-Base Formula.
Hint: Begin from bx = c and start by applying logd( ) to both sides.
6. Write these special case versions of the change-of-base formula using the simplest notation
available.
a. case where d = 10
b. case where d = e ≈ 2.71828
Homework:
Do section 3.4 exercises 23-29 odd, 31-36, 39, 41, 44, 45, 55, 56, 60, 62.
Hints:
• “common logarithm” means base 10 logarithm, written log10( ) or log( )
• “natural logarithm” means base e logarithm, written loge( ) or ln( )
• For 39 and 41, use change-of-base formula to write each function in terms of ln( )’s.