help me with LOGARITHM!!?

Question

Can u prove that the logarithm of the power of a number is equal to the exponent times the logarithm of the number?

Answer 1

Sure. Suppose ln x=y. Then e^y=x. Therefore, (e^y)^n=x^n By exponent laws, (e^y)^n=e^(yn) Therefore, e^(yn)=x^n. Therefore, ln (x^n)=yn. But y=ln x. Thus by substitution, ln (x^n)=n ln x. Q.E.D.

Answer 2

I can give you another.

This is a protocol:
log(p,m) <> means as logarithm of p in base of m.

y = log (x^n,a)
=> a^y=x^n
=> x=a^(y/n)
==make as logarithm=> log(x,a) = y/n
==multiple to n bothsides => n*log(x,a)=y
==replace y as we defined first => n*log(x,a) = log(x^n,a)

you can write the last line as this:
log(x^n,a)=log(x,a) + log(x,a) + .... + log(x,a) (n Times)

Be Succeed.

Answer 3

LOG[ a^m] = LOG[a*a*a*......m times]
= LOGa + LOGa +LOG a+ ......... m times
= m times LOGa
= m*LOGa
Hence proved

Answer 4

log(x^n)= log(x*x*x*x*x*x*x*............n times)
= logx + logx + logx + logx + logx +.........n times
= n*( logx )

you should have figured this out.