Logarithms as the Inverse of Exponentials
Read logₐ x as the answer to the question 'what power of a gives x'. The log graph is the exponential graph reflected in y = x, and each function undoes the other.
What a learner can do afterwards
- Rewrite 2⁵ = 32 in logarithm form and back again
- Explain why the logarithm of a negative number has no value
- Sketch y = ln x by reflecting y = eˣ in the line y = x
1 · Read
Read log base a of x as a question: what power of a gives x. So log base 2 of 32 is 5, because 2 to the 5 is 32. Power form and log form are the same fact twice. Fluency both ways turns log equations into plain power questions.
Some questions have no answer. The log of zero or a negative has no value, because no power of a positive base lands there. Since e to the x only outputs positives, ln x only accepts positives. Domain limits are built in.
Sketch y equals ln x by reflecting y equals e to the x in the line y equals x. Where the exponential crosses (0, 1), the log crosses (1, 0). Where the exponential soars, the log climbs slowly. The curve hugs the y axis, showing the domain edge.
Exponentials and logs undo each other by swapping inputs and outputs. One takes a power and gives a number, the other returns the power. That swap is exactly why their graphs mirror. Name which direction each function runs before solving.
A log names the missing power, accepts positives only, and mirrors its exponential.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.