log(xy)=log x+log y || log(ab)=loga+logb proof || lnxy)=lnx+lny || log(m+n)=logm+logn
log(xy)=log x+log y, log(ab)=loga+logb proof. How to prove the product rule of a logarithm? If not, then this video is for you.
You might be knowing that log-base-b of a is defined when b not equal to one and a and be both positive real numbers. log(m+n)=logm+logn.
Let x and y be two real numbers;
When the product of x and y is negative; then we have to prove,
Log-base-b-of absolute value of x times y; equals the sum of log-base-b of the absolute value of x and log-base-b of the absolute value of y.
OR,
When the product of x and y is positive; then we have to prove,
Log-base-b-of x times y; equals the sum of log-base-b of the absolute value of x and log-base-b of the absolute value of y.
Let log-base-b of x equals m; this implies b to mth power equals x.
and log-base-b of y equals n; this implies b to nth power equals y.
Let us evaluate the product of x and y. Which is equal to the product of b to m and b to the n, which is same as b to the m plus n.
Taking log to base b, on both sides of this equation; we get;
Log of x times y equals log of b to the m+n power.
Since, the expression on right-hand-side of this equation becomes, m plus n; this implies, log of x times y equals m plus n.
Now, simply substitute the value of m and n, to get,
Log of x times y equals the sum of log of x and log of y.
Please note that log of x plus y is not equal to the sum of log of x and log of y, and log of x upon y is not equal to log of x upon log of y.
So, what next? In the next video, I will prove the quotient rule of log.
If you enjoyed watching this video, please hit the like button and consider subscribing my channel to get upcoming uploads. Thank You.
Keywords:
log(xy)=log x+log y,log(xy),log(xy)=,lnxy)=lnx+lny,ln(xy)=x+y,ln(xy),ln(x+y)=ln(x)+ln(y),log(a+b) formula,log(a+b)=loga+logb,log(a+b)=log(ab),log(a+b) equals,log a base b,log(x+y)=logx+logy,log(x+y)=logx+logy then x=,log(x+y) rule,log(x+y)=,log(x+y) simplify
