1. Base Stays Base: The small number at the bottom of the Exponent ($b^x$) is the same small number at the bottom of the Log ($\log_b y$).
2. The Question: When you see $\log$, ask: "Base to the what power gives me the big number?"
3. The Inverse: $b^x = y \iff \log_b y = x$.
4. The Common Log: If you see $\log$ with no base, it means base 10.
Rewrite each expression in its opposite form.
From Power to Root: $2^5 = 32$.
From Power to Root: $10^4 = 10,000$.
From Root to Power: $\log_3(81) = 4$.
From Root to Power: $\log_5(125) = 3$.
Solve these logarithms without a calculator by "counting" the multiplications.
$\log_2(64) = $
$\log_{10}(1,000,000) = $
$\log_4(16) = $
$\log_7(49) = $
What is the value of $\log_b(1)$? It doesn't matter what the base is! Why is the answer always the same?
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(Hint: Any base to the power of WHAT equals 1?)
Remember: Each whole number on the Richter scale is 10 times more powerful than the one before.
The Quake: If an earthquake is a level 8, and another is a level 5... how many times more powerful is the level 8?
The Sound: The decibel scale for sound is also logarithmic. A whisper is 20 dB. A conversation is 60 dB. How many "powers of 10" louder is the conversation?
What happens if the result is a fraction?
Example: $\log_{10}(0.1) = -1$.
Because $10^{-1} = 1/10 = 0.1$.
Task: Solve these "Downwards" Logs:
Objective: Teach a younger student how to "compress" big numbers.
The Activity: Write down these numbers: 10, 100, 1,000, 10,000, 100,000.
The Question: "Instead of saying 'one hundred thousand,' can you just tell me how many zeros it has? (5)."
The Lesson: "In high school math, we call that a 'Logarithm.' A log is just a way to say how many times we multiplied by 10. It makes big numbers much easier to talk about."
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