Logarithms are the inverse of indices. They help us answer questions like '2 to what power gives 32?' The answer is 5, because 2^5 = 32. Logarithms make it easier to work with very large or very small numbers.
▸ Definition of a Logarithm — If a^x = N (where a > 0, a ≠ 1, N > 0), then log_a(N) = x. This is read as 'log base a of N equals x'.
Example: Since 2^5 = 32, we can write this in logarithmic form as log_2(32) = 5.
◆ Diagram: Table showing Exponential Form and Logarithmic Form:
Exponential Form | Logarithmic Form | Example
a^x = N | log_a(N) = x | 2^5 = 32 → log_2(32) = 5
10^3 = 1000 | log(1000) = 3 | Common log (base 10 assumed)
e^1 = e | ln(e) = 1 | Natural log (base e ≈ 2.718)
★ Real-world — Earthquake Magnitude:
The Richter scale is logarithmic. A magnitude 7 earthquake is 1,000 times more intense than a magnitude 4 earthquake because 10^(7-4) = 10^3 = 1,000. Logarithms compress huge ranges of intensity into manageable numbers.
Audio narration — reads aloud, works offline
In your own words, summarise the main idea of this lesson.