Hello, my name is Angel, and I am here to talk about logarithms and what they are used for.
A logarithm is the inverse of an exponent.
So basically, “what exponent can I raise this number to get this value?”.
This is what it looks like.
How this works is this little part...is your base.
The number next to your log...is the result of applying a power to that base.
And your answer...is that power.
When you rearrange it...
...it starts to become a bit more clear on what you got to do!
This is called exponential form.
To solve, you just need to remember your powers.
If you can’t remember a whole power, or one cannot be found, then you would have to change base, which is when you put the log of your answer over the log of your base, like this.
And when I mean "answer", I mean the result of applying that power to the base.
When changing base, it is best to use a calculator to find your (log) value.
Quick example: log base 3 of 27 is equal to what?
If you can do it on your own, pause and solve it yourself.
If not, stick around for a bit for the process.
...
Done?
You should have gotten 3.
If you reorganize the log into exponential form, you would get 3 to the power of x is equal to 27.
3^3 is 27, so your answer would be 3.
Yeah, you could do change of base like this:
...but it would take longer to do so and would require a calculator.
It is best to use this method if you know that the value of the power is not going to be a whole number, such as log base 5 of 21 is equal to x...
...which would equal about 1.89 if you do the change of base.
There is also expanding and condensing logarithms, which I’ll quickly touch on.
Just to quickly note, the base must stay the same the entire way through, and you cannot condense or expand logarithms with different bases.
Big no-no.
If a logarithm is set up like this:
...then it equals to log base b of x plus log base b of y, and vice versa.
If a logarithm is set up like this:
...then it equals to log base b of x minus log base b of y, and vice versa.
And if a logarithm is set up like this:
...then it is equal to the log of base b of x to the power of a.
Remember, you are applying a to the x only if you change it to this method.
But if you keep the *former form, you multiply a to your entire log, which is cleaner in my opinion, once you do find the logarithm that is.
These can be done in combination with each other, like this:
and in this case you would just do this:
Apply a to all parts of the ratio, then expand it to the quantity of log base b of x to the power of a plus log base b of y to the power of a minus...
oh look blooper
...log base b of z to the power of a.
And this is just one...
oh look another blooper
And this is just one combination you can be met with.
Again, this works the other way around.
For a more in depth talk about condensing and expanding, click the video in the description by Joe Schiavone
because I only have so much time to animate and I still need to get to my to-
These are used in the real world, quite often actually. A good example is decibels.
Sound is measured in decibels, along with power level and voltage.
The “-bel” in “decibel” comes from the Bell Telephone Labs after Alexander Graham Bell, the group that...
i am an animator, not a talker
...created this system.
while “deci-” means one tenth.
That means that one decibel means a 1/10 change in sound, voltage, etc.
But from this point on, I’ll only be talking about sound intensity.
Decibel values depend on how far, loud, and fast a sound is travelling.
The equation used to calculate differences in decibels is:
tfw you forget to remove a pause in post
P is equal to power, with P1 representing your reference sound power, and P2 representing the actual power compared to your reference.
And your answer is how loud a sound is in the current situation.
This also...
i sound 12
Keep in mind this is also important.
Because when a question gives you the decibel, this is basically (a part of) your answer.
Unless there's more than one thing going on, which I'll explain later.
For calculating from threshold sound, you can just use:
...where I is equal to how many times louder a sound is than the threshold.
Technically, the equation is 10 log (I/I0), where that I0 means your threshold sound...
but I becomes how many times louder a sound is times I0, as in I becomes 2I0, 3I0, etc.; so that I0 is basically cancelled out.
Threshold sound is basically the quietest sound an organism perceives.
Since humans have differing hearing set ups compared to a dog, cat, bird, whatever, we typically use humans...
...because, y'know, we're humans.
tbh at this point i don't remember if this gap was intentional.
But basically, in a nutshell, this tells you how many decibels louder is it from threshold sound.
Although you do not need this value, but is important nonetheless, the threshold sound for humans is typically 4 dB, which is **4 time quieter than breathing at 10 dB.
Yes, there are sites that list average decibel values, such as BYJU’S chart here:
...but it is also important to calculate when multiple sounds are playing simultaneously, or you only know how much louder a sound is from the threshold.
And most chart values are only averages or approximates and depend on how close and *fast they are as well.
Here’s a problem:
Let’s say you have a couple of air horns. These air horns are about 130 decibels at 2 feet away. About how loud are they?
