Solving exponential equations

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

Now you ask the other way round: for which \(x\) does \(y\) have a certain value? You solve equations like \(2^x = 20\) by trying, with the same base and with the logarithm.

📄 Open worksheet (PDF, German)

The video

In the video you look for the matching \(x\) on the graph. Then you calculate it with the logarithm.

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

The most important points

Trying

\(2^x = 8\): \(2^1 = 2\), \(2^2 = 4\), \(2^3 = 8\). So \(x = 3\).

On the graph: from y = 8 horizontally to the curve, then vertically down to x = 3. — On the graph: from \(y = 8\) horizontally to the curve, then vertically down to \(x = 3\).

Same base

Write both sides as a power of the same base: \(3^x = 81 = 3^4\), so \(x = 4\). \(2^x = \frac{1}{4} = 2^{-2}\), so \(x = -2\).

Logarithm

\(2^x = 20\) cannot be found by trying. It holds \(x = \dfrac{\lg 20}{\lg 2} \approx 4{,}32\). Check: \(2^{4{,}32} \approx 20\).

2^x = 20 has the solution x \approx 4{,}32. — \(2^x = 20\) has the solution \(x \approx 4{,}32\).

Application: interest

\(1000\) € at \(5\,\%\): \(1000 \cdot 1{,}05^x = 2000\), so \(1{,}05^x = 2\) and \(x = \dfrac{\lg 2}{\lg 1{,}05} \approx 14{,}2\) years.

When do 1000 € become two thousand euros? — When do \(1000\) € become two thousand euros?

Careful: \(2^x = 20\) does not mean \(x = 20 : 2 = 10\). Because \(2^{10} = 1024\). The \(x\) is in the exponent, you cannot simply divide. Use the 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)

Show the worksheet solutions

Try it yourself

Task

Solve \(3^x = 50\) with the logarithm.

Show solution

\(x = \dfrac{\lg 50}{\lg 3} \approx 3{,}56\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Exponentialgleichungen lösen (Probieren, gleiche Basis, Logarithmus). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

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.