Logarithms
ICSE Class 9 · Mathematics · Chapter 8
What this chapter covers
- From indices to logarithms
- Exponential form and logarithmic form
- Evaluating logarithms and finding unknowns
- Conditions and special results
- Common logarithms
- The product law
- The quotient law
- The power law
- Using the laws together
- Expanding expressions with the laws of logarithms
- Writing an expression as a single logarithm
- Solving equations involving logarithms
Key points
- ay = x and loga x = y say the same thing; a logarithm is an index.
- The base must be positive and not 1; the argument must be positive.
- loga 1 = 0, loga a = 1, loga an = n and aloga x = x.
- log x with no base written means the common logarithm, to the base 10.
- Product law: loga (mn) = loga m + loga n.
- Quotient law: loga ((m)/(n)) = loga m - loga n, and loga (1)/(n) = -loga n.
- Power law: loga mn = n loga m, so log √(m) = (1)/(2) log m.
- No law splits log (m + n) or log (m - n), and (log m)/(log n) is not log (m)/(n).
- To expand, apply the quotient law, then the product law, then the power law: y = (a4 b2)/(c3) gives log y = 4 log a + 2 log b - 3 log c.
- To combine, move multipliers up as indices first, write lone numbers as logarithms, then collect into one logarithm.
- In an equation, reject any value that makes an argument zero or negative.
Questions answered
Define the logarithm of a number to a given base, and write log7 343 = 3 in exponential form.
The logarithm of a positive number x to a base a is the index to which a must be raised to give x, where a is positive and not 1. In exponential form, 73 = 343.
Write in logarithmic form: (a) 3-4 = (1)/(81), (b) 64(1)/(3) = 4.
(a) log3 (1)/(81) = -4. (b) log64 4 = (1)/(3).
Evaluate log4 32.
Let 4y = 32. Then 22y = 25, so 2y = 5 and y = (5)/(2).
Find x if logx 0.001 = -3.
x-3 = 0.001 = (1)/(1000), so x3 = 1000 and x = 10.
Explain why loga 1 = 0 for every allowed base a.
In exponential form the statement is a0 = 1, which is true for every non-zero a. The power of a that gives 1 is therefore 0.
Explain why log10 (-5) has no value.
It would need a power of 10 equal to -5. Every power of a positive base is positive, so no such power exists.
Using the laws of indices, explain why the product law of logarithms is true.
If m = ax and n = ay, then mn = ax+y by the product law of indices. So loga mn = x + y = loga m + loga n. Adding logarithms is adding indices.
Explain why loga mn = n loga m.
If m = ax, then mn = (ax)n = anx, so loga mn = nx = n loga m. The power law of indices multiplies the index by n.
Past board exam questions
ICSE 2019 sample paper · 3 marks
If log4 = 0.6020, find the value of log80.
Show answer
log 4 = 2 log 2 = 0.6020 ⇒ log 2 = 0.3010.
log 80 = log (8 × 10) = 3 log 2 + log 10 = 3 × 0.3010 + 1 = 1.9030.