Here are the slides on the problems we did today.
I saw the most trouble with change of base. Let me give you the way to think about it one more time:
Say we are converting from base-2 to base-10. Look (click) at the diagram below and it should be clear that the unit distance (the distance between 0 and 1) on the base-10 scale is about 3 times as much as the unit distance on the base-2 scale.
So, if we knew exactly how many base-2 units it took to make a base-10 unit then we would have a conversion factor. Look closely again at the diagram. You should see that the base-10 scale is exactly `(log_10(10))/(log_10(2))` times as big. This, then is our conversion factor.
If the bases were equal our conversion would be `log_2(x)=log_10(x)`.
They are not equal so we apply the conversion factor and get: `log_2(x)= (log_10(10))/(log_10(2))*log_10(x)`.
This simplifies to give us the typical change of base formula: `log_2(x)=(log_10(x))/(log_10(2))`
This argument can be applied to the change of any base to another.
Here is a second area of trouble: Everyone is comfortable with `10^x=10000`. I think everyone is also ok with `log_10(10000)=x`. The first equation asks “What is the exponent I raise 10 to in order to get 10000″? The second equation asks the same question right? So when you see `log_3(x+1)=4` the question is “What is the exponent I raise 3 to in order to get `x+1`”? Write the question as an exponential equation: `3^a=(x+1)`. You know the answer because the logarithmic equation has a number after the equal sign. The logarithm(base 3 of `x+1`)is the exponent(4). Therefore, `3^4=x+1`.