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The logarithm

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

Two to the power of what is eight? The logarithm looks for the exponent: \( \log_2 8 = 3 \), because \( 2^3 = 8 \).
📄 Open worksheet (PDF, German)

The video

Watch the colours: the base is blue, the exponent is orange, the number is purple.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:16 The notation
  3. 0:42 Examples
  4. 1:01 The common logarithm
  5. 1:18 Laws of logarithms
  6. 1:57 Drawing graphs
  7. 2:18 One base is enough
  8. 2:57 Checking with the graph
  9. 3:21 History
  10. 3:38 Try it yourself

The most important points

Logarithm

\( b^x = z \iff \log_b z = x \). Question: base to the power of what gives the number?

Power: result wanted. Root: base wanted. Logarithm: exponent wanted.
Power: result wanted. Root: base wanted. Logarithm: exponent wanted.

\( \log_2 16 = 4 \), \( \log_3 81 = 4 \). Base 10: \( \lg 1000 = 3 \), \( \lg 0.01 = -2 \).

Graphs of \log_2 x , \log_3 x and \lg x : all pass through (1 \,|\, 0) .
Graphs of \( \log_2 x \), \( \log_3 x \) and \( \lg x \): all pass through \( (1 \,|\, 0) \).

Special values: \( \log_b 1 = 0 \), \( \log_b b = 1 \). Application: \( 100 \cdot 2^x = 1600 \Rightarrow 2^x = 16 \Rightarrow x = 4 \).

Laws of logarithms

\( \lg(a \cdot c) = \lg a + \lg c \) and \( \lg(a^n) = n \cdot \lg a \). It follows: \( 2^x = 10 \Rightarrow x \cdot \lg 2 = \lg 10 \Rightarrow x = \frac{\lg 10}{\lg 2} \approx 3.32 \).

Change of base: \log_b z = \lg z/\lg b – with the common logarithm you can calculate every logarithm.
Change of base: \( \log_b z = \frac{\lg z}{\lg b} \) – with the common logarithm you can calculate every logarithm.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

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Show the worksheet solutions
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Try it yourself

Task

Calculate: a) \( \log_2 32 \) b) \( \lg 10\,000 \) c) \( \log_5 25 \) d) \( \log_3 1 \)

Show solution

a) \( 5 \) b) \( 4 \) c) \( 2 \) d) \( 0 \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: the logarithm (base to the power of what gives the number, for example log 2 of 8 = 3, common logarithm). Give me five short tasks, one after the other. Always wait for my answer. Then tell me if it is right and explain briefly why. Write in simple German and put an English translation in brackets after each sentence.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.