Logarithm and Anti-Logarithm Operations: Videos & Practice Problems

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 explore the fundamental concepts of logarithms and their applications, particularly in the context of chemistry. A key concept is the pH scale, which is defined as the negative logarithm of the hydronium ion concentration, represented as pH = -\log[H_3O^+] or pH = -\log[H^+]. Understanding logarithms is essential before delving into the specifics of acids and bases.

When working with logarithmic calculations, it's important to distinguish between the characteristic and the mantissa of a number. For example, in the number 12.005, the characteristic is the part to the left of the decimal point (12), while the mantissa is the part to the right (0.005). This distinction is crucial because the number of digits in the mantissa of your logarithmic result corresponds to the number of significant figures in the original measurement.

As you practice calculating logarithms, remember that the significant figures in your answer are determined by the mantissa. For instance, if you take the logarithm of a number with three significant figures, your answer should also reflect three digits in the mantissa. This principle will guide you in solving logarithmic problems accurately.

To reinforce your understanding, try calculating the logarithm of various values and determine the final answers based on the significant figures. If you encounter difficulties, further resources are available to guide you through specific examples.

Log and Anti-Log

Understanding significant figures is crucial when performing logarithmic and anti-logarithmic calculations. The number of digits in the mantissa of your answer corresponds directly to the number of significant figures in the original problem. For instance, when calculating the logarithm of a number expressed in scientific notation, such as \(1.15 \times 10^{-5}\), it is essential to recognize that this number has three significant figures. Consequently, when you compute the logarithm, you should ensure that the mantissa of your answer also contains three digits. Thus, the logarithm yields approximately \(-4.9393\), which should be rounded to \(-4.939\) to maintain the correct number of significant figures.

In another example, when calculating the logarithm of \(100\), which is expressed as \(1.00 \times 10^{2}\), the presence of a decimal point indicates that it also has three significant figures. Therefore, the logarithm of \(100\) results in \(2.000\), ensuring that the answer reflects the same number of significant figures as the original number.

When dealing with anti-logarithms, the process is somewhat reversed. The anti-logarithm of a number \(x\) is defined as \(10^x\). Here, the number of significant figures in the result is determined by the number of significant figures in the mantissa of the original logarithmic value. This means that if the mantissa of the logarithmic value has three significant figures, the anti-logarithm will also reflect three significant figures in its answer.

In summary, whether calculating logarithms or anti-logarithms, always ensure that the number of significant figures in your final answer corresponds to the significant figures present in the original number. This practice is essential for maintaining precision in scientific calculations.

Log and Anti-Log

When calculating the antilogarithm of a number, it is essential to understand the concept of significant figures, particularly focusing on the mantissa, which is the part of the number after the decimal point. The number of significant figures in the final answer corresponds to the number of digits in the mantissa of the original number.

For example, when taking the antilog of -4.18, the mantissa is 0.18, which contains 2 digits. Therefore, the final answer must also have 2 significant figures. The antilog can be expressed mathematically as:

\[\text{antilog}(x) = 10^x\]

Calculating the antilog of -4.18 gives:

\[\text{antilog}(-4.18) = 10^{-4.18} \approx 6.6069 \times 10^{-5}\]

However, since we need to round this to 2 significant figures, the final answer is:

\[6.6 \times 10^{-5}\]

In another example, consider the expression \(10^{0.0033}\). Here, the mantissa is 0.0033, which has 4 digits. Thus, the final answer must have 4 significant figures. Performing the calculation yields:

\[10^{0.0033} \approx 1.0076\]

Rounding this to 4 significant figures results in:

\[1.008\]

Understanding the distinction between the characteristic and mantissa is crucial, especially in analytical chemistry, where precision in calculations, including uncertainties, is vital. This knowledge will significantly aid in achieving accurate results in various calculations.

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.