If c>0, then the following equality holds,
the integral from a to b dx/x equals to the integral from ac to bc dx/x.
Show your work and an explanation for each step.
"\\displaystyle\\int_a^b \\frac{dx}{x}=\\int _{ac}^{bc}\\frac{dx}{x}" ;
let's integrate the left and right parts of the equality:
"\\displaystyle\\ln |x||_a^b =\\displaystyle\\ln |x||_{ac}^{bc}" ;
"\\displaystyle\\ln |b|-\\ln|a| =\\displaystyle\\ln |bc|-\\ln|ac|" ;
according to the property of the logarithm we can write the following formula:
"\\displaystyle\\ln\\left|\\frac ba\\right|=\\ln\\left|\\frac{bc}{ac}\\right|" ;
"\\displaystyle\\ln\\left|\\frac ba\\right|=\\ln\\left|\\frac ba\\right|" - the identity is proved.
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