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Grade 11 · Logarithms and domains

Logarithms and domains: The domain starts at one

Logarithms and domains: investigate the domain starts at one with domain boundary a = 1; logarithm base b = 2.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Logarithms and domains: The domain starts at one. Allowed input x: 1.2. Logarithm value: -2.322. Base raised to output: 0.2A positive argument has a real logarithm-2.322.321.23.66yx → · labeled axes rescale to this model
Only bases greater than one are allowed, avoiding the invalid bases zero and one. The trace never touches the domain boundary. Logarithms are evaluated with natural logs via the change-of-base identity.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Allowed input x
1.2
Logarithm value
-2.322
Base raised to output
0.2

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Understand what you are seeing

The idea behind the motion.

A real logarithm requires a positive argument. The shift a moves that boundary, while a base greater than one controls how quickly output grows. Exponentiating the output recovers the positive argument, not the original x until the shift is added back. This investigation starts with Domain boundary a = 1; Logarithm base b = 2. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

y=log_b(x−a); bʸ=x−a; x>a

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    State the domain x>a before looking for values or inverse formulas. The starting case is “The domain starts at one.”

  2. STEP 2

    Follow the changing quantity

    Move the input through positive argument values from 0.2 to 5. Notice that arguments below one give negative logarithms.

  3. STEP 3

    Explain and test the result

    Check the inverse relation b raised to the output equals x−a. Vary the base without changing the argument and explain the change in output.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Domain boundary a = 1; Logarithm base b = 2. Pause the timeline at 80%. Given allowed input x = 5.04, calculate logarithm value, base raised to output. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

The argument is 5.04−(1)=4.04. Change of base gives ln(4.04)/ln(2)=2.014355; raising 2 to this output recovers 4.04. Results: Logarithm value: 2.014; Base raised to output: 4.04. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 11 · Algebra

Combine logarithms without admitting forbidden inputs

Preserve each logarithm's positivity requirement when solving.