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Learn · Algebra 2

Logarithmic Functions: The Inverse of Exponential

A logarithm asks "what exponent?" Master the definition, the three properties (product/quotient/power), and you will solve exponential equations of any base.

9 minTEKS 5A,5B,5CAlgebra 2

A log answers one question: "what exponent?"

logb(x) asks: "to what power do I raise b to get x?" That's it. Once you internalize this question, every log problem becomes a translation exercise.

logb(x) = y ⟺ by = xA log equation and an exponential equation say the same thing two different ways.
Logarithm y = log₂(x): passes through (1, 0), (2, 1), (4, 2)V.A. x = 0(1, 0)(2, 1)(4, 2)Inverse of y = 2ˣ · reflection across y = x · domain (0, ∞), range (−∞, ∞)

Reading logs naturally

log2(8) = 3 (2 to what power = 8? Answer: 3)log10(1000) = 3 (10 cubed = 1000)log2(32) = 5 (2⁵ = 32)ln(e²) = 2 (ln means log base e)
Read a log directly
Evaluate the logarithm log₂(32) (base 2).

The three properties

Product: log(ab) = log(a) + log(b)Quotient: log(a/b) = log(a) − log(b)Power: log(an) = n · log(a)All three convert multiplication/division/exponents into addition/subtraction/multiplication. That's the magic of logs.

Combining logs

log(8) + log(125)= log(8 · 125)= log(1000)= 3 (assuming base 10)
Combine using product property
Use log properties to simplify: log(8) + log(125).
Change of base: log_b(x) = log(x) / log(b) or ln(x) / ln(b)log_b(x) =ln(x)ln(b)Calculator uselog₇(50) = ln(50)/ln(7)= 3.912/1.946 ≈ 2.01

Change of base formula

Most calculators only have log10 (just "log") and ln (log base e). To compute log3(20), use:

logb(x) = log(x) / log(b) = ln(x) / ln(b)log3(20) = log(20) / log(3) ≈ 1.301 / 0.477 ≈ 2.727

Logs and exponentials are inverses

The undo trick

To solve bx = y, take logb of both sides: x = logb(y). To solve logb(x) = y, raise b to both sides: x = by.

Domain check

logb(x) is only defined for x > 0. After solving a log equation, always check that your answer makes the original argument positive — otherwise it's extraneous.

Log scale examples: Richter, pH, decibels — each unit = 10× change45678felt indoorsshakingdamagemajorcatastrophicRichter magnitudeeach +1 = 10× amplitude, ~32× energyM 8 releases ~1 million× energy of M 4

3-second recap

  • logb(x) = y means by = x.
  • Three properties: product → sum, quotient → difference, power → multiplier.
  • Change of base: logb(x) = log(x)/log(b).
  • Logs and exponentials undo each other.

Check yourself

Quick check #1
Evaluate log₂(8).
Quick check #2
Solve for x: log(x) = 2 (assume base 10).