Notes · Unit 2
Logarithms
Twelve short slides on scientific notation, logarithms, their laws, change of base and common mistakes.
Press Next (or the right arrow key) to move on. On most slides, each press reveals one more idea.
Scientific notationDefinitionCharacteristic and mantissaLawsChange of base
1 · Scientific notation
Write big and small numbers neatly
d × 10n
1 ≤ d < 10n is an integer
Big number
4500 = 4.5 × 103 point moves 3 places left
4500 = 4.5 × 103 point moves 3 places left
Small number
0.00045 = 4.5 × 10−4 point moves 4 places right, so the exponent is negative
0.00045 = 4.5 × 10−4 point moves 4 places right, so the exponent is negative
Not scientific notation
12.5 × 103 has d = 12.5, which is too big. Write 1.25 × 104.
12.5 × 103 has d = 12.5, which is too big. Write 1.25 × 104.
2 · Logarithms
A logarithm is an exponent
logb x = y ⇔ by = x
Read it as: the power to which b must be raised to get x is y.
log2 8 = 3 because 23 = 8
log10 1000 = 3 because 103 = 1000
log9 3 = ½ because 91/2 = 3
3 · Rules for a valid logarithm
logb x needs three conditions
b > 0b ≠ 1x > 0
Why not a negative number?
log5(−1) would need 5y = −1, but a positive base never gives a negative result.
log5(−1) would need 5y = −1, but a positive base never gives a negative result.
Why not base 1?
log1 2 would need 1y = 2, but 1 raised to any power is 1.
log1 2 would need 1y = 2, but 1 raised to any power is 1.
Two values to remember
logb 1 = 0 since b0 = 1
logb b = 1 since b1 = b
logb 1 = 0 since b0 = 1
logb b = 1 since b1 = b
Example: log2(4 − 2x) is undefined at x = 2, because the bracket becomes 0.
4 · Two special bases
Common, natural and antilog
Common logarithm
Base 10
Written log x. Also called Briggsian logarithms.
Natural logarithm
Base e
Written ln x, where e ≈ 2.71828 (irrational). ln e = 1 and ln 1 = 0.
Antilogarithm
The antilog undoes the log: if log x = y, then x = antilog y.
The antilog undoes the log: if log x = y, then x = antilog y.
5 · Parts of a common logarithm
Characteristic and mantissa
log x = characteristic + mantissa
Characteristic is the integral part. If x = d × 10n, it equals n. For 0.04 = 4 × 10−2 it is −2.
Mantissa is the fractional part. It is never negative and is always less than 1.
Bar notation. log 0.0123 = 2.0899 means −2 + 0.0899. Only the 2 is negative; the mantissa stays positive.
Counting digits. A whole number has characteristic + 1 digits. For example, 1009 has characteristic 3 and 4 digits.
6 · Laws of logarithms
Three laws turn hard sums into easy ones
Product law
logb(mn) = logb m + logb n
logb(mn) = logb m + logb n
Quotient law
logb(m/n) = logb m − logb n
logb(m/n) = logb m − logb n
Power law
logb(mn) = n logb m
logb(mn) = n logb m
Example. Given log 2 = 0.3010 and log 3 = 0.4771, find log 12.
12 = 22 × 3, so log 12 = 2(0.3010) + 0.4771 = 1.0791
12 = 22 × 3, so log 12 = 2(0.3010) + 0.4771 = 1.0791
Single logarithm. 5 log 2 − 2 log 5 = log 32 − log 25 = log(32/25)
7 · Change of base
Move a logarithm to a new base
logb m = loga mloga b
Use a calculator. log7 20 = log 20 ÷ log 7, using base 10 for both.
Reciprocal rule. loga b = 1 ÷ logb a, so the two multiply to 1.
Example. log2 3 × log3 4 = (log 3 ÷ log 2) × (log 4 ÷ log 3) = log2 4 = 2
8 · Using logarithms
Where logarithms help
How many digits in 220?
log 220 = 20 × 0.3010 = 6.020. The characteristic is 6, so the number has 7 digits.
log 220 = 20 × 0.3010 = 6.020. The characteristic is 6, so the number has 7 digits.
Richter scale. M = log(I ÷ I0). If the intensity equals I0, then M = log 1 = 0.
Comparing earthquakes. Each extra unit multiplies intensity by 10. Magnitude 7 against 5 is a difference of 2, so 102 = 100 times stronger.
9 · Common mistakes
Five traps to avoid
Splitting a sum. log(50 − 5) ≠ log 50 − log 5. The laws handle products and quotients only, never sums or differences inside the bracket.
Confusing a quotient. log10 x ÷ log10 y is logy x, not log(x/y).
Skipping the check. After solving log(x + 3) = log(15x − 4) you get x = 0.5. Both brackets must be positive: each equals 3.5, so it is valid.
Wrong sign for a base. If log√x 25 = 4, then x = 5 or −5, but a base must be positive, so x = 5.
Positive exponent for small numbers. 0.04 = 4 × 10−2, so its characteristic is −2, not 2.
10 · Quick check
Try these, then tap to check
11 · Your turn
Practise Unit 2
Three timed MCQ tests with an explanation after every answer.
Practice questions are independently written and are not official FBISE papers.