Logarithm Notation

Direct Answer

The notation log_b(x) = y means that b^y = x. Here, b is the base (a positive number ≠ 1), x is the argument (> 0), and y is the exponent to which the base must be raised to yield x.

Informal Interpretation

A logarithm answers the question “How many of one number (the base) multiply together to make another number (the argument)?” For instance, log₁₀(1000) = 3 because you multiply 10 by itself 3 times to get 1000.

Formal Definition

For a base b satisfying b > 0 and b ≠ 1, the function log_b is the inverse of the exponential function b^x:

If b^y = x, then log_b(x) = y.

Notational Variants

• Common (decimal) logarithm: log₁₀(x), often written log(x) in engineering.
• Natural logarithm: ln(x) ≡ log_e(x), where e ≈ 2.71828.
• Binary logarithm: log₂(x), common in computer science.

Fundamental Properties

• Product rule: log_b(xy) = log_b(x) + log_b(y)
• Quotient rule: log_b(x/y) = log_b(x) – log_b(y)
• Power rule: log_b(x^k) = k·log_b(x)
• Change of base: log_b(x) = log_k(x)/log_k(b) for any valid base k.

Domain Restrictions

• The argument x must satisfy x > 0.
• The base b must satisfy b > 0 and b ≠ 1.
• Logarithms are undefined for nonpositive arguments in the real number system.

Examples

1. log₂(8) = 3 because 2³ = 8.
2. ln(e⁵) = 5 because e⁵ = e⁵.
3. log₁₀(0.01) = –2 because 10^–2 = 0.01.
4. log₅(25³) = 3·log₅(25) = 3·2 = 6.

Common Misconceptions

• log_b(x + y) does not equal log_b(x) + log_b(y).
• Forgetting the base leads to ambiguity; log(x) must be interpreted by context.

Next Steps

1. Explore solving exponential and logarithmic equations.
2. Investigate how logarithms linearize multiplicative relationships.
3. Apply logs in data analysis: decibels, Richter scale, pH scale.
4. Extend to complex logarithms and understand multi-valued branches.