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Exponential and logarithmic graphs
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Exponential and logarithmic functions undo each other. Knowing the shape of their graphs makes the algebra much easier to picture.
The graph of y = ax
For a > 1, y = ax:
- passes through (0, 1), because a0 = 1
- is always positive
- gets closer and closer to the x-axis on the left: the x-axis (y = 0) is an asymptote
- increases more and more steeply on the right.
- passes through (0, 1), because a0 = 1
- is always positive
- gets closer and closer to the x-axis on the left: the x-axis (y = 0) is an asymptote
- increases more and more steeply on the right.
What a logarithm means
logb x asks: what power of b gives x?
logb x = y means exactly the same as by = x.
log2 8 = 3 because 23 = 8.
log10 1000 = 3 because 103 = 1000.
logb x = y means exactly the same as by = x.
log2 8 = 3 because 23 = 8.
log10 1000 = 3 because 103 = 1000.
The graph of y = logb x
For b > 1, y = logb x:
- passes through (1, 0), because b0 = 1
- only exists for x > 0
- has the y-axis (x = 0) as an asymptote
- is the reflection of y = bx in the line y = x.
- passes through (1, 0), because b0 = 1
- only exists for x > 0
- has the y-axis (x = 0) as an asymptote
- is the reflection of y = bx in the line y = x.
Find log3 81.
- Ask: 3 to what power gives 81?
- 34 = 81
Answer: log3 81 = 4
logb x = y means by = x. y = ax passes through (0, 1) with asymptote y = 0. y = logb x passes through (1, 0) with asymptote x = 0, and is the reflection of y = bx in y = x.
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