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.
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.
More to read
6.3 Logarithmic Functions
A logarithm asks: to which power must the base go to make this number? So log base 2 of 32 is 5, because 2^5 = 32. Converting between power form and log form is the same fact written two ways. Fluency both ways turns scary log equations into plain power questions.
6.4 Graphs of Logarithmic Functions
The graph of y = ln x is the mirror of y = e^x in the line y = x. Where e^x crosses (0, 1), ln x crosses (1, 0); where e^x soars, ln x climbs slowly. The curve hugs the y-axis on one side because the log of zero or negatives has no value.
1.4 Inverse Functions
Inverse functions undo each other by swapping inputs and outputs. Exponentials take a power and give a number; logarithms take the number and return the power. Their graphs mirror in the line y = x for exactly this reason. Since e^x only outputs positives, ln x only accepts positives.
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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.
14 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.