Below if the video transcript of my tutorial video MCAT Math Without A Calculator Video 8 – Logarithms and Negative Logs.
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 https://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 (2.3×10^-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.
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Watch This video Here: MCAT Math Without A Calculator Video 8 – Logarithms and Negative Logs