Transcript Definition of Logarithmic Function

4.3 – Logarithmic Functions
Definition of Logarithmic Function
A logarithm is defined as follows:
𝑙𝑜𝑔𝑎 𝑥 = 𝑦 represents 𝑎 𝑦 = 𝑥, where 𝑎 > 0, and 𝑎 ≠ 1
The number 𝑙𝑜𝑔𝑎 𝑥 is the power y to which we raise a to get x. The
number a is called the logarithmic base.
𝑙𝑜𝑔𝑎 𝑥 is read as “the logarithm, base a, of x.”
The domain is the set of all positive real numbers 0, ∞ .
The range is the set of all real numbers −∞, ∞ .
The x-intercept is 1 and the y-intercept does not exist.
The y-axis (x = 0) is a vertical asymptote.
4.3 – Logarithmic Functions
The graph of the logarithmic function is shown below.
𝑎, 1
𝑦 = 𝑙𝑜𝑔𝑎 𝑥
𝑎
The common logarithmic function
𝑦 = 𝑙𝑜𝑔10 𝑥 = 𝑙𝑜𝑔 𝑥
𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓
𝑥 = 10𝑦
𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓
𝑥 = 𝑒𝑦
The natural logarithmic function
𝑦 = 𝑙𝑜𝑔𝑒 𝑥 = ln 𝑥
4.3 – Logarithmic Functions
𝑮𝒓𝒂𝒑𝒉𝒔 𝒐𝒇 𝑳𝒐𝒈𝒂𝒓𝒊𝒕𝒉𝒎𝒊𝒄 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔
𝑦 = 𝑙𝑜𝑔2 𝑥
𝑦 = ln 𝑥
𝑦 = log 𝑥
4.3 – Logarithmic Functions
Graphs of 𝑓 𝑥 = 𝑙𝑜𝑔𝑎 𝑥 𝑎𝑛𝑑 𝑓 𝑥 = 𝑎 𝑥
𝑦 = 𝑓 𝑥 = 𝑎𝑥
a
𝑎, 1
𝑦 = 𝑙𝑜𝑔𝑎 𝑥
𝑎
Inverse Functions: 𝑓 𝑥 = 𝑙𝑜𝑔𝑎 𝑥 𝑎𝑛𝑑 𝑓 𝑥 = 𝑎 𝑥
4.3 – Logarithmic Functions
Graphs of 𝑓 𝑥 = 𝑒 𝑥 𝑎𝑛𝑑 𝑓 𝑥 = ln 𝑥
Inverse Functions: 𝑓 𝑥 = 𝑒 𝑥 𝑎𝑛𝑑 𝑓 𝑥 = ln 𝑥
4.3 – Logarithmic Functions
𝑙𝑜𝑔𝑎 1 = 0
𝑙𝑜𝑔𝑎 𝑎𝑟 = 𝑟
𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔 𝒐𝒇 𝑳𝒐𝒈𝒂𝒓𝒊𝒕𝒉𝒎𝒔
𝑙𝑜𝑔𝑎 𝑎 = 1
𝑎𝑙𝑜𝑔𝑎 𝑀 = M
𝑙𝑜𝑔𝑎 𝑀𝑟 = 𝑟𝑙𝑜𝑔𝑎 𝑀
𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔 𝒐𝒇 𝑵𝒂𝒕𝒖𝒓𝒂𝒍 𝑳𝒐𝒈𝒂𝒓𝒊𝒕𝒉𝒎𝒔
𝐥𝐧 𝐱 = 𝒍𝒐𝒈𝒆 𝒙
ln 1 = 0
ln e = 1
ln 𝑒 𝑟 = 𝑟
ln 𝑀𝑟 = 𝑟lnM
𝑒 𝑙𝑛𝑘 = k
4.3 – Logarithmic Functions
𝑾𝒓𝒊𝒕𝒆 𝒂𝒔 𝒆𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝒆𝒙𝒑𝒐𝒏𝒆𝒏𝒕𝒊𝒂𝒍 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔
𝑙𝑜𝑔3 𝑥 = 5
𝑙𝑜𝑔2 𝑥 = 8
𝑥 = 35
𝑥 = 28
𝑙𝑜𝑔256 4 = 14
4=
ln 𝑎 = 6
𝑎 = 𝑒6
1
2564
𝑾𝒓𝒊𝒕𝒆 𝒂𝒔 𝒆𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝒍𝒐𝒈𝒂𝒓𝒊𝒕𝒉𝒎𝒊𝒄 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔
8𝑥 = 4
53 = 125
𝑀7 = R
𝑙𝑜𝑔8 8𝑥 = 𝑙𝑜𝑔8 4
𝑙𝑜𝑔5 53 = 𝑙𝑜𝑔5 125
𝑙𝑜𝑔𝑀 𝑀7 = 𝑙𝑜𝑔𝑀 R
𝑥𝑙𝑜𝑔8 8 = 𝑙𝑜𝑔8 4
3𝑙𝑜𝑔5 5 = 𝑙𝑜𝑔5 125
7𝑙𝑜𝑔𝑀 𝑀 = 𝑙𝑜𝑔𝑀 R
𝑥 = 𝑙𝑜𝑔8 4
3 = 𝑙𝑜𝑔5 125
7 = 𝑙𝑜𝑔𝑀 R
4.3 – Logarithmic Functions
𝑮𝒓𝒂𝒑𝒉𝒊𝒏𝒈 𝑳𝒐𝒈𝒂𝒓𝒊𝒕𝒉𝒎𝒊𝒄 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔
𝑦 = 𝑙𝑜𝑔2 𝑥
𝐷𝑜𝑚𝑎𝑖𝑛:
(0, ∞)
𝑅𝑎𝑛𝑔𝑒:
(−∞, ∞)
𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒𝑠:
𝑉𝐴 @ 𝑥 = 0
𝑦 = 𝑙𝑜𝑔2 𝑥 − 1
𝐷𝑜𝑚𝑎𝑖𝑛:
(0, ∞)
𝑅𝑎𝑛𝑔𝑒:
(−∞, ∞)
𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒𝑠:
𝑉𝐴 @ 𝑥 = 0
𝑦 = 𝑙𝑜𝑔2 (𝑥 − 2) − 1
𝐷𝑜𝑚𝑎𝑖𝑛:
(2, ∞)
𝑅𝑎𝑛𝑔𝑒:
(−∞, ∞)
𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒𝑠:
𝑉𝐴 @ 𝑥 = 2
4.3 – Logarithmic Functions
𝑨𝒑𝒑𝒍𝒊𝒄𝒂𝒕𝒊𝒊𝒐𝒏𝒔 𝒐𝒇 𝑳𝒐𝒈𝒂𝒓𝒊𝒕𝒉𝒎𝒊𝒄 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔
The pH scale is used to represent the level of acidity or alkalinity of a liquid. A pH of
less than 7 is acidic and a pH greater than 7 is alkaline (basic). It is based on the molar
concentration of hydrogen ions, 𝐻+ . The following formula is used to calculate the
pH of any liquid based on the measure of the hydrogen ions:
𝑝𝐻 = −𝑙𝑜𝑔 𝐻 + .
Bleach and milk of magnesia are bases. Their hydrogen ion values are 2.0 × 10−13 and
4.1 × 10−10 , respectively.
What is the pH of bleach?
What is the pH of milk of magnesia?
𝑝𝐻 = −𝑙𝑜𝑔 𝐻+
𝑝𝐻 = −𝑙𝑜𝑔 𝐻+
𝑝𝐻 = −log(2.0 × 10−13 )
𝑝𝐻 = −log(4.1 × 10−10 )
𝑝𝐻 = 12.7
𝑝𝐻 = 9.4
What liquid is more basic?
bleach