Logarithm and Anti-Logarithm Operations - Online Tutor, Practice Problems & Exam Prep

Log and Anti-Log

In this section, we'll take a look at the calculations involved in logarithms as well as anti-log functions. Now, we're going to say in Gen Chem, we learned that pH just represented the negative log of the concentration of our hydronium ion, which could be written as H₃O⁺ or H⁺. Now, eventually, we'll get to that portion again when we're talking about determining the pH of acidic and basic solutions. But realize here, we're going to be introduced to logarithms right now before we even get to acids and bases.

Now we're going to say when dealing with a number with a decimal point like the one that we have below of 12.005, we're going to say the portion to the left of the decimal point, which is right here, we refer to that as the characteristic. And then we're going to say the portion to the right of the decimal point, this portion here, this is referred to as our mantissa. Knowing that will be incredibly important when we take a look at log functions as well as anti-log functions.

We're going to say, when taking the log of a number, the number of digits in the mantissa of your answer is equal to the number of significant figures in the problem. Knowing this, look to see if you could answer the example that's right below where I'm giving you the log of different values and from that, determine what the final answer will be. Try and attempt this on your own. If you get stuck, don't worry. Just click over to the next video. You'll see how I approach those 2 examples that are found in example 1. Give it a try and then come back and see how I do it.

Log and Anti-Log

So here we need to provide the answer with the correct number of significant figures for each of the following. All right. So going back to what it says here, it says the number of digits in the mantissa of your answer is equal to the number of significant figures in the problem. If we take a look here at the first one we have log1.15×10-5, we're going to say here that our value has 3 significant figures in it. So there are 3 significant figures in the problem. Therefore, the number of digits in the mantissa also has to be 3. So that's how it works. So if you take your calculator, type in log1.15×10-5, you'll get back negative 4.9393. But we want our answer to have 3 digits in the mantissa. So that's the portion after our decimal point. So the correct answer would be negative 4.939. So that would be our answer at the end.

For the next one, we have log100. Here we have a decimal point so remember we're going this way. Our first non-zero number is this 1 and if you count all the way through, we also have 3 significant figures there. So that means our answer, once we find the log100, must have 3 digits after the decimal point. So if you plug that in, you'll get 2.000 as your correct answer.

Now that we've done that, let's look to see if we can do the antilogarithm portion. You're going to say here, when we're dealing with antilogarithms, we're going to say the antilog of a number x is equal to 10x. And we're going to say when taking the antilog of a number, we're going to say the number of significant figures in the answer is equal to the number of significant figures in the mantissa of the problem. Okay. So think of this working in the opposite direction. So look at how many digits you have in the mantissa of the problem being asked. That will help you determine the number of significant figures you'll have in your answer. So give this a try. If you get stuck, don't worry. Just come back. We'll take a look at it together and see how we get to the answers. So good luck, guys.

Log and Anti-Log

So here it says provide the answer with the correct number of significant figures for each of the following. So let's reread what it set up above. It said here, when taking the anti-log of a number, the number of significant figures in the answer is equal to the number of digits in the mantissa of the problem. Remember, the mantissa is the portion after the decimal point. If we take a look here, we want to take the anti-log of negative 4.18. Here, the mantissa is the portion after the decimal point. We have 2 digits there in the mantissa. So that means our answer has to have 2 significant figures. So remember, antilog really just means 10 to that number. If we punch that into our calculators, we will get 6.6069-5. But again remember, the number of digits in our mantissa is 2 which means we need to have 2 sig figs for our final answer. So this will come out to 6.6-5 as our final answer.

Then here, it's already written in the way that we need it. It's 100.0033. Here, the number of digits in the mantissa is 4. So that means our answer at the end must have 4 sig figs. So when we punch that into our calculator, we'll have 1.00763. But, again, we need 4 sig figs for the final answer. So that comes out to 1.007 as the final answer to this question.

Just remember the terms of characteristic versus mantissa and the operations that we do whether we're taking log of that or the antilog of this. It will become incredibly important as we delve deeper into analytical chemistry when we start talking about other things such as uncertainties and other types of calculations where this little bit of information will go a long way to determining the final answer.