A2M3L13­SB Changing the Base.notebook
February 10, 2016
Changing the Base of Logs
(also includes natural logs and applying all properties of common logs to logs of any base)
We have mainly been working with logs with base 10 but not all logs are base 10. Since we have a log key on the calculator, it is convenient to change logs of any base to base 10.
We have mainly been working with logs with base 10 but not all logs are base 10. Since we have a log key on the calculator, it is convenient to change logs of any base to base 10.
A2M3L13­SB Changing the Base.notebook
February 10, 2016
Change of Base Formula for Logarithms
To change a log expression in base a to an equivalent log expression in base b:
loga(x)
logb(x) = loga(b)
x, a, and b are all postive real numbers with a ≠ 1 and b ≠ 1.
log(x)
logb(x) = log(b)
x, a, and b are all postive real numbers with a ≠ 1 and b ≠ 1.
A2M3L13­SB Changing the Base.notebook
February 10, 2016
A2M3L13­SB Changing the Base.notebook
February 10, 2016
Find the approximate value of the following log expressions to 4 decimal places.
1) log3(32)
2) log2(32)
3) log5(120)
4) log8(334) A2M3L13­SB Changing the Base.notebook
February 10, 2016
Properties
base 10 logs
base b logs
log (10k x) = k+log(x)
The Natural Logarithm
le logarithme naturel (LN)
Another commonly used base for logarithms is Euler's number, e. loge(x) = ln(x)
frequently used to model growth and change in the natural world
all properties of logarithms of other bases also apply to natural logaritms
values are obtained by using the ln key on the calculatoor
A2M3L13­SB Changing the Base.notebook
February 10, 2016
Determine the value of the following without the use of a calculator:
Rewrite as an equivalent expression containing only one logarithm.
1) 2ln(x)­3ln(y)
1
2) 3ln(a)­ ln(b)
2
a
3) ln( )­ln(ab)
b
A2M3L13­SB Changing the Base.notebook
Given
and
approximate the value of ln(8/3) without a calculator.
approximate the value of ln(108) without a calculator.
Homework: Worksheet
February 10, 2016
A2M3L13­SB Changing the Base.notebook
February 10, 2016