Question

I need help with logarithims. MEDAL and FAN. Question below...

Answer 1

\[WhatPower _{100}(0.01)=?\]

Answer 2

What power? what?

Answer 3

WhatPower is another word for log

Answer 4

oh that's clever hehe

Answer 5

We have a few steps that will head us in the right direction.\[\large\rm \log_{100}\left(0.01\right)\quad=\log_{100}\left(\frac{1}{100}\right)\]Are you ok with this one?
Converting our decimal to a fraction.

Answer 6

Yup I'm okay with it

Answer 7

Next we'll apply an exponent rule:\[\rm \frac{1}{x}=x^{-1}\]

Answer 8

Do you see how we can apply this rule to the 1/100? :)

Answer 9

Not really

Answer 10

Following the rule,\[\rm \frac{1}{100}=100^{-1}\]we can bring the 100 out of the denominator by putting a -1 power on it.

Answer 11

\[\large\rm \log_{100}\left(\frac{1}{100}\right)\quad=\log_{100}\left(100^{-1}\right)\]

Answer 12

You're probably more comfortable with `positive exponents`.
Positive exponents tell us that we're `multiplying` stuff together.
Example: \(\rm x^4=x\cdot x\cdot x\cdot x\)

Well `negative exponents` work exactly the same but in the opposite way.
Negative exponents tell us that we're `dividing` by a bunch of stuff.
Example: \(\rm x^{-4}=\dfrac{1}{x\cdot x\cdot x\cdot x}\)

Answer 13

So we're applying this idea in reverse,
in our fraction, we're dividing by 100,
so we can rewrite it with this negative exponent which represents division.

Answer 14

Okay I get it @zepdrix

Answer 15

Sorry about leaving all of a sudden, I had to go to lunch

Answer 16

From this point:\[\large\rm \log_{100}\left(\frac{1}{100}\right)\quad=\log_{100}\left(100^{-1}\right)\]Apply a log rule:\[\rm \log(a^b)=b\cdot \log(a)\]

Answer 17

Do you see how we can apply this rule to the -1?

Answer 18

yes

Answer 19

\[\large\rm \log_{100}\left(100^{-1}\right)\quad=-1\cdot\log_{100}\left(100\right)\]So that let's us bring the -1 out front, ya? :)

Answer 20

yep

Answer 21

Another log rule,
we'll switch back to your log wording so this makes sense.\[\large\rm what~power_{100}(100)\]100 to which power will give us 100?

Answer 22

1

Answer 23

Good good good,\[\large\rm -1\cdot\color{orangered}{\log_{100}\left(100\right)}\quad=-1\cdot\color{orangered}{1}\]

Answer 24

Whenever the base of the log matches the contents of the log,
the result is just 1.

Answer 25

So looks like we end up with -1, ya? :)

yayyy team!

Answer 26

Could you answer some more of these type of questions?

Answer 27

btw thank you so much for the help