The math behind percentage change

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Key facts: The log of a product is the sum of the logs.
tex2html_wrap_inline168 for small tex2html_wrap_inline170 .
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Take a log (natural) regression.
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tex2html_wrap_inline172
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Increase x by tex2html_wrap_inline170 percent, how does ln(y) shift?
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tex2html_wrap_inline178
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tex2html_wrap_inline180
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tex2html_wrap_inline182
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How much did ln(y) shift?
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tex2html_wrap_inline186 .
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tex2html_wrap_inline188 .
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tex2html_wrap_inline190 .
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tex2html_wrap_inline192 .
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tex2html_wrap_inline194 .
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Finally: percentage change in y is tex2html_wrap_inline196 .