Logarithms: how many times to multiply
A logarithm asks how many times you multiply the base to reach a number. Change the base and two numbers, and see on a slide-rule scale that multiplying numbers adds their logarithms.
Core concept
log_b x is the power of b that gives x (b^y = x means log_b x = y); multiplying numbers adds their logarithms: log (xy) = log x + log y
Topic: Indices and logarithms
Indices tell you how many times to multiply: 2 × 2 × 2 = 2³ = 8. A logarithm asks the same question the other way round: how many times must you multiply 2 to get 8? The answer is 3, so log₂ 8 = 3. In general, if bʸ = x then log_b x = y, where b is the base. Base 10 logarithms are the common ones: log 1000 = 3, because 10³ = 1000. Logarithms turn multiplication into addition: since 2³ × 2² = 2⁵, log₂ 8 + log₂ 4 = log₂ 32. This is the product rule, log (xy) = log x + log y. A slide rule uses it: two scales marked in logarithms are laid end to end, so adding lengths multiplies numbers.
What you will learn
- Explain a logarithm as the power of the base that gives a number.
- Find logarithms to base 2 and base 10 of simple numbers.
- Use the product rule log (xy) = log x + log y.
- Recognise the quotient and power rules of logarithms.
Before you start
- Know the laws of indices.
Materials
- A notebook for a table of powers of 2 and 10
Equipment
- A calculator with a log key (optional)
Procedure
- With base 2, record x = 8 and y = 4.
- Change to base 10, set x = 1000 and y = 100, and record.
- With base 10, set x = 10 and y = 10, and record.
How this simulation is simplified
- Logarithms are rounded to three decimal places.