Logarithm and Anti-Logarithm Operations: Study with Video Lessons, Practice Problems & Examples

When calculating logarithms, the number of significant figures in the result corresponds to the significant figures in the original number. For example, the log of 1.15 × 10^-5 yields -4.939, maintaining three significant figures. Conversely, when finding the anti-logarithm, the significant figures in the answer depend on the mantissa of the input. Thus, understanding significant figures is crucial for accurate calculations in chemistry, particularly in reactions and titrations.

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, in Gen Chem, we learned that pH represents the negative log of the concentration of our hydronium ion, which could be written as H₃O⁺ or H⁺. Eventually, we'll revisit this concept when talking about determining the pH of acidic and basic solutions. But first, we're going to introduce logarithms before we even get to acids and bases.

Now, when dealing with a number with a decimal point like the one below of 12.005, we refer to the portion to the left of the decimal point as the characteristic.

Then, the portion to the right of the decimal point 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 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, try to see if you can answer the example 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 where you'll see how I approach those two examples.

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 log of 1.15 times 10 to the negative 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 and type in log of this number, 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'd be our answer at the end. For the next one, we have a log of 100. 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 log of this number, must have 3 digits after the decimal point. So if you plug that in, you will get 2.000 as your correct answer.

So remember, when taking the log of a number, the number of significant figures in that number is equal to the number of digits in your mantissa. All right. Now that we've done that, let's look to see if we can do the anti-logarithm portion. You're going to say here, when we're dealing with anti-logarithms, we're going to say the anti-log of a number x is equal to 10 to the x. And we're going to say when taking the anti-log 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 said above. It said here, when taking the antilog 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 antilog of -4.18.

Here, the mantissa is the portion after the decimal point. We have 2 digits there in the mantissa. That means our answer has to have 2 significant figures. Remember, antilog really just means \(10\) to that number. And if we punch that into our calculators, we get \(6.6069 \times 10^{-5}\).

But again, remember the number of digits in our mantissa is 2, which means we need to have 2 significant figures for our final answer. So this will come out to \(6.6 \times 10^{-5}\) as our final answer. Then here, it's already written in the way that we need it. It's \(10^{0.003}\). Here, the number of digits in the mantissa is 3.

So that means our answer at the end must have 3 significant figures. So when we punch that into our calculator, we'll have 1.0076. But again, we need 3 significant figures for the final answer. 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 the 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.

Here’s what students ask on this topic:

How do you determine the number of significant figures in the result of a logarithm calculation?

When calculating the logarithm of a number, the number of significant figures in the result corresponds to the number of significant figures in the original number. For example, if you take the log of 1.15 × 10^-5, which has three significant figures, the result should also have three significant figures in the mantissa (the portion after the decimal point). Therefore, the log of 1.15 × 10^-5 is -4.939, maintaining three significant figures in the mantissa.

What is the relationship between significant figures in the mantissa and the original number when calculating logarithms?

The number of significant figures in the mantissa of the logarithm result is equal to the number of significant figures in the original number. For instance, if the original number has three significant figures, the mantissa of the logarithm result should also have three significant figures. This ensures that the precision of the original number is accurately reflected in the logarithm calculation.

How do you calculate the anti-logarithm of a number and determine the significant figures in the result?

To calculate the anti-logarithm of a number, you use the formula 10^x, where x is the given number. The number of significant figures in the result is determined by the number of significant figures in the mantissa of the input number. For example, if the mantissa of the input number has three significant figures, the result of the anti-logarithm should also have three significant figures. This ensures that the precision of the input number is accurately reflected in the anti-logarithm calculation.

Why is understanding significant figures important in logarithm and anti-logarithm calculations in chemistry?

Understanding significant figures is crucial in logarithm and anti-logarithm calculations in chemistry because it ensures the accuracy and precision of the results. Significant figures reflect the precision of the measurements and calculations, which is essential in chemical reactions and titrations. Accurate significant figures help avoid errors and ensure that the results are reliable and meaningful in scientific experiments and analyses.

Can you provide an example of calculating the logarithm and anti-logarithm with significant figures?

Sure! Let's calculate the logarithm of 1.15 × 10^-5. The original number has three significant figures. Using a calculator, log(1.15 × 10^-5) = -4.9393. Since the original number has three significant figures, the mantissa of the result should also have three significant figures, so the answer is -4.939. Now, let's calculate the anti-logarithm of -4.939. Using the formula 10^x, we get 10^-4.939 = 1.15 × 10^-5. The result has three significant figures, matching the precision of the input mantissa.