Inverse of y = ln(x)
Q01. Diagram
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Q02. Inverse of y = ln(x)
Inverse of y = ln(x)
- Inverse of y = ln(x) is y = e^x
- Inverse of y = e^x is y = ln(x)
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Q03. Properties
- 1. ln(e^x) = x
- This the law of logarithm
- 2. e^(ln(x)) = x
- Let e^(ln(x)) = y
- Take logarithm on both sides
- ln(e^(ln(x)) = ln(y)
- ln(x) = ln(y)
- Hence y = x
- Hence e^(ln(x)) = x
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Q04. Find y'
First derivative of e^x
- y = e^x and y' = e^x
- Since e^x is greater than 0
- Hence the curve is always increasing
- y = ln(x) and y' = 1/x
- Since x is always greater than 0
- Hence the curve is always increasing
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Q04. Find y"
Second derivative of e^x
- y = e^x and y" = e^x
- Since e^x is greater than 0
- Hence the curve is always concave upward
- y = ln(x) and y" = -1/(x^2)
- Since x is always greater than 0
- Hence the curve is always concave downward
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