1. ## PROVE LOG PROPERTY!

Can someone help me with these questions?

I need to prove each property by using the format like this:
Change of Base
Let y=LOGa X
1.then a to the power of y=x definition of logarithm.
2.LOGb a to the power of y=LOGb X take LOGb of both sides
3.y times LOGb a=LOGb X log of a power
4.y= LOGb X / LOGb a, QED divide by LOGb a

CAN SOMEONE HELP ME WITH THE FOLLWOING QES?

Direct Proportionality
LOGa=(c)(LOGb X),where c is a constant.

Product of two logs
(LOGa B)(LOGb C)=LOGa C

log with a power for its base
LOG(b to the power n) X=1/n LOGb X

base and argument are reciprocals
LOG1/b 1/X =LOGb X

I NEED IT FOR TOMORROW!!
THANK YOU SO MUCH!!

2. Originally Posted by kellyliu
Can someone help me with these questions?

I need to prove each property by using the format like this:
Change of Base
Let y=LOGa X
1.then a to the power of y=x definition of logarithm.
2.LOGb a to the power of y=LOGb X take LOGb of both sides
3.y times LOGb a=LOGb X log of a power
4.y= LOGb X / LOGb a, QED divide by LOGb a

CAN SOMEONE HELP ME WITH THE FOLLWOING QES?

Direct Proportionality
LOGa=(c)(LOGb X),where c is a constant.

A) Product of two logs
(LOGa B)(LOGb C)=LOGa C

B) log with a power for its base
LOG(b to the power n) X=1/n LOGb X

C) base and argument are reciprocals
LOG1/b 1/X =LOGb X

I NEED IT FOR TOMORROW!!
THANK YOU SO MUCH!!
To A):
Obviously B = b

$\displaystyle \log_a(b) \cdot \log_b(c)=\dfrac{\log(b)}{\log(a)} \cdot \dfrac{\log(c)}{\log(b)}=\dfrac{\log(c)}{\log(a)}= \log_a(c)$

to B):

$\displaystyle \log_{b^n}(X)=\dfrac{\log(X)}{\log(b^n)}=\dfrac{\l og(X)}{n \log(b)}=\dfrac1n \cdot \dfrac{\log(X)}{\log(b)}=\dfrac1n \cdot \log_b(X)$

to C):

$\displaystyle \log_{\frac1b}\left(\dfrac1x\right)=\dfrac{\log\le ft(\dfrac1x\right)}{\log\left(\dfrac1b\right)} = \dfrac{-\log(x)}{-\log(b)}=\dfrac{\log(x)}{\log(b)}=\log_b(x)$

3. thank you sooo much!!!