Leah here from Leah4sci.com/MCAT and in this
video, I'll show you how to tackle logarithms
style tackled questions without a calculator
as it may show up on your MCAT.
Logs are tricky and unlike other questions
that you can fallback on the long mathematical
process like Multiplication, Division, Addition,
Subtraction, with Logs, you either know it
or you don't! So I wanna make sure that if
you're ask to calculate anything like a Ph
or a Pka on your MCAT, you know what to do
and you know how to do it quickly and confidently.
And the good news is that on the MCAT, when
you're ask to solve a log type of question,
the answer choices will be so far apart that
if you quickly get a value that is close enough,
you're good enough to have the answer and
you're able to move on and devote more time
to your next question. You may find yourself
faced with the question that reads
Find the pH of a 2.3 x 10¯⁴ molar solution
of NaOH.
The Chemistry portion of this question will
be tackled in my Chem videos at http://leah4sci.com/CHEMISTRY
but for now let's focus on Math. To find the
pH of the solution we first have to see the
Ion concentration given. Since we're given
an NaOH or OH minus concentration, we'll go
for the minus OH concentration to the pOH,
from there we'll get to the pH because pH
plus pOH is fourteen.
But the difficult part is going from the OH
minus concentration to the pOH. We'll use
the following formula: pOH is equal to negative
log of the OH minus concentration which in
this question is negative log two point three
times ten to the minus 4 (-log (2.3x10^-4).
Great! I have the setup but I don't have the
calculator, how do I proceed?
So how do we find the negative log of this
value? Let's focus on the trick first and
then we'll come back to solve this question.
Here's what you have to know about logs on
the MCAT. When given log of a number, you're
really given log base ten and this tells you,
ten to the what power is equals to this number.
Don't worry, we're not going to the long tedious
solving of logs, instead we'll understand
what's going on and then I'll show you a quick
shortcut.
This is easy for simple questions, for example,
given log a hundred, ask yourself, ten raised
to what power is equal to a hundred (100)?
Since ten times ten or ten squared is a hundred,
the answer is 2. What about the log of a thousand
(1000)? Ten to the what power is equal to
one thousand. Well ten times ten times ten
or ten to third equals to a thousand and so
the answer is 3.
Now what about the negative logs?While negative
logs appear to be more difficult its actually
the same thing with a negative before the
power. So for example; negative log of zero
point one is equal to one because one times
ten to the minus one is equal to zero point
one.
Another way to think of this is turn this
into Scientific Notation. Zero point one is
really one times ten to the minus one so we
already have ten raised to a power of a negative
number and here's the trick; when your number
is written out in Scientific Notation, you
have ten to the negative number. Just circle
that number and that is your answer.
Let's try this shortcut again; say you're
told negative log of one times ten to the
minus four. We have one times ten to a negative
power, circle that power and the answer is
four. What about negative log of one times
ten to minus eleven? Again, ten to the negative
number, circle that number and the answer
is eleven. Let's apply this to something you
already know.
A neutral solution at room temperature will
have a pH at seven. This solution will also
have an equivalent H plus and OH minus concentration
each equaling one times ten to the minus seven.
So let's prove this mathematically. The pH
is equal to negative log, the H plus concentration
which is equal to negative log one times ten
to the minus seven. We have our value set
up at one times ten to a negative power, grab
that power and the answer is seven. But what
if we don't have one times ten to a power
and instead have two point three times ten
to a power such as the example given.
This is where we are going to apply additional
tricks and simplification keeping in mind
that we only need an answer that is close
enough. Let's start with something simple;
say we're trying to solve the negative log
of four point five times ten to the minus
three. First we wanna get a ballpark but we
need this expression to be setup as one times
ten to a negative power so I wanna find the
in-between values by taking four point five
rounding it down to one and up to ten because
this way my number will start with the one.
If I round it down to one, I get one times
ten to the minus three and if I round it up
to ten I get ten times ten to the minus three.
But ten times ten to a power is not a proper
Scientific Notation so we want to divide the
ten by ten giving me one and then multiply
the exponent by ten or raise it by one power
giving me times ten to the minus two. Ten
times ten to the minus three is the same thing
of saying one times ten to the minus two.
Now I can solve for the negative logs, one
times ten to the minus three gives me three,
one times ten to the minus two gives me two.
And so this was an H plus concentration, my
pH would be somewhere between two and three.
This is where you wanna be careful. Four point
five seems like the halfway between looks
like four and five but it's not going to be
the halfway between my pHs. This is where
you have to make a decision, check your MCAT
choices, if you see only one value with the
pH two and three pick that answer and save
your time.
But if you have multiple values you need to
zero in a little more we can take this trick
a step further. What you want to understand
moving forward is that a logarithmic scale
is going to go ten to a hundred thousand so
we'd start small and then it sky rockets so
the numbers are going to follow a similar
pattern.
In addition to name the values between one
and ten, it helps to recognize if not to memorize
what the negative log values will be if your
coefficient is a three, a five, a seven or
an eight because then you can always extract
like the numbers in between.
I'll show you the calculator values and then
I'll show you the numbers that you want to
study. The negative log of one times ten to
the minus three is three; we already know
that. But as we increase from one times ten
to the minus three all the way to ten times
ten to the minus three, we're moving closer
and closer to two. So as your coefficient
goes up, your pH, your pKa or the number that
we derive is going to go down.
Three times ten to the minus three in the
calculator is equal to two point fifty three.
What I want you to recognize is three times
any power is going to give me a number point
five. Five times ten to the minus three on
the calculator is two point three, so recognize
that if we have a five we get a number point
3. Eight times ten to the minus three is equal
to two point one so any number starting with
an eight will give a number point one.
And obviously if we add in ten times ten to
the minus three which is the same thing as
saying one times ten to the minus two, our
answer is going to be two. So here's the pattern
to recognize, three starts with the five that's
like your halfway point and then we go down
three down to one we're at the lower number.
In other words, for the quicker version, if
you're given five times ten to the minus three,
we recognize the number is near three but
because it's a five another one we have to
go down and the answer will be two point something.
In this case two point three.
So let's go back to our initial example, we're
trying to solve two point three times ten
to the minus four. First we want to find the
range and the range will be, this is my lowest
number because I round two point three down
to one. One times ten to the minus four is
four. Rounding it up to ten, two point three
becomes ten times ten to the minus four or
one times ten to the minus three giving me
three.
So we know the pOH is going to be between
three and four. Remember the trick when we
use three times ten to the minus three and
it was two point five three? In this case,
two point three is very close to three so
the answer should be very close to point five.
But we have to take the smaller number, it
won't be four point five, it will be something
like three point five because we know our
pOH range has to be between three and four.
Three point five in the MCAT will be close
enough and then we do fourteen minus three
point five which is our pOH and that gives
us an answer of ten point five for our pH.
Punching this example into the calculator
gave me an answer of ten point three six which
on the MCAT is close enough.
Be sure to join me in the next video where
I show you a similar shortcut how to solve
antilogs or when you're given a pH pOH or
pKa and have to find the concentration or
the ka value.
Are you stuck on a specific MCAT topic? I
offer Private Online Tutoring where I focus
on your needs to strengthen your individual
weaknesses. Tutoring details can be found
using the link below or by visiting my website
leah4sci.com/MCATTutor.
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