Học · 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 phút TEKS 5A,5B,5C Algebra 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 = x A log equation and an exponential equation say the same thing two different ways.

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 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.

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).