Common and Natural Logarithms
Grade 10 Algebra — Mastering lg, ln, and the Change of Base Formula
1. The Common Logarithm (lg)
Definition: The common logarithm is the logarithm with base 10. Instead of writing log₁₀b, we write lg b.
On a scientific calculator, the common logarithm is computed using the lg or log key.
Key Examples:
lg 100 = 2
since 10² = 100
lg 0.001 = −3
since 10⁻³ = 0.001
lg 10 = 1
since 10¹ = 10
lg 1 = 0
since 10⁰ = 1
In general: lg b = x means 10ˣ = b.
2. The Natural Logarithm (ln)
Definition: The natural logarithm is the logarithm with base $e$, where $e \approx 2.7182818\ldots$ is an irrational number. Instead of writing loge b, we write ln b.
On a scientific calculator, the natural logarithm is computed using the ln key.
Key Examples:
ln e = 1
since e¹ = e
ln 1 = 0
since e⁰ = 1
ln e³ = 3
since e³ = e³