LOG PROPERITIES
(a, b, M , N  0, b, B  1)
CONCEPTS
𝒃𝒙 = 𝒂

𝐥𝐨𝐠 𝒃 𝒂 = 𝒙
EXAMPLES
𝐥𝐨𝐠 𝒃 𝑴𝑵 = 𝐥𝐨𝐠 𝒃 𝑴 + 𝐥𝐨𝐠 𝒃 𝑵
𝑴
𝐥𝐨𝐠 𝒃 = 𝐥𝐨𝐠 𝒃 𝑴 − 𝐥𝐨𝐠 𝒃 𝑵
𝑵
𝐥𝐨𝐠 𝟑 𝟗 = 𝟐
log 4 x  log 4  log x
2
2
2
x2
)  log x 2  log y
10 y
10
10
log (
log 25x  x log 25
𝐥𝐨𝐠 𝒃 𝑴𝑵 = 𝑵 𝐥𝐨𝐠 𝒃 𝑴
logb M 
𝟑𝟐 = 𝟗

5
log B M
log B b
5
log 5 
3
log e 5 ln 5

log e 3 ln 3
Note:
log 2 x  x
log bk  k
2
b
b0  1

log1  ln1  0
log 1  0
b
log x means log
10
log 5  log10 5
x
ln 3  loge 3
ln x means log e x
Also Note:
You CANNOT take the log of a negative number…
log
10
 7   undefined
log  M  N   log M  log N
“log” is NOT distributive…
log MN   log M  log N 
log 
N
M
Learning Center (CC-315)
  log M

 log N