The Logarithm Laws : The Logarithm Laws
Slide 2: Since a logarithm is simply an exponent which is just being written down on the line, we expect the logarithm laws to work the same as the rules for exponents.
Slide 3: Exponents Logarithms
bm × bn = bm+n logb xy = logb x + logb y
bm ÷ bn = bm-n logb (x/y) = logb x − logb y
(bm)n = bmn logb (xn) = n logb x
b1 = b logb (b) = 1
b0 = 1 logb (1) = 0
Slide 4: Note: On our calculators, "log" (without any base) is taken to mean "log base 10". So, for example "log 7" means "log107".
1. Expand log 7x as the sum of 2 logarithms. : 1. Expand log 7x as the sum of 2 logarithms. Sol: Using the first law given above, our answer is
log 7x = log 7 + log x
Note: This has the same meaning as 107 × 10x = 107+ x
2. Using your calculator, show thatlog (20/5) = log 20 − log 5. : 2. Using your calculator, show thatlog (20/5) = log 20 − log 5. using numbers this time so you can convince yourself that the log law works.
LHS
= log (20/5)
= log 4
= 0.60206 (using calculator)
Now,
Slide 7: RHS
= log 20 − log 5
= 1.30103 − 0.69897 (using calculator)
= 0.60206
= LHS
We have shown that the second logaritm law above works for our number example.
3. Express as a multiple of logarithms: log x5. : 3. Express as a multiple of logarithms: log x5. Using the third logarithm law, we have
log x5 = 5 log x
We have expressed it as a multiple of logarithm, and it no longer involves an exponent.
Slide 9: Note 1: Each of the following is equal to 1:
log6 6 = log10 10 = logx x = loga a = 1
The equivalent statements, using ordinary exponents, are as follows:
61 = 6
101 = 10
x1 = x
a1 = a
Slide 10: Note 2: All of these are equivalent to 0:
log7 1 = log10 1 = loge1 = logx 1 = 0
The equivalent statments in exponential form are:
70 = 1
100 = 1
e0 = 1
x0 = 1
THANK YOU : THANK YOU