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Chapter 10: Problem 11

Find each logarithm. Give approximations to four decimal places. \(\log 0.0326\)

Short Answer

Expert verified
\( \log 0.0326 \approx -1.4863 \)

Step by step solution

01

- Understand the Logarithm

The logarithm asks what power the base (usually 10 for common logarithms) must be raised to obtain the given number. For this problem, the base is 10, and the number is 0.0326.
02

- Use a Calculator

To find \( \log 0.0326 \), input 0.0326 into a scientific calculator and press the log button. Scientific calculators often have dedicated buttons for logarithmic functions.
03

- Read the Result

The calculator will display an approximation of the logarithm. For \( \log 0.0326 \), the calculator gives approximately -1.4863.
04

- Round to Four Decimal Places

Make sure the result from the calculator is rounded to four decimal places. The value -1.48627 rounds to -1.4863 at four decimal places.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithms
Logarithms are a mathematical concept that help us understand the power to which we must raise a base number to get another number. In most common logarithms, we use base 10. For example, \(\text{log}_{10}(100) = 2\) because 10 raised to the power of 2 equals 100. Think of logarithms as the inverse function of exponentiation. If exponentiation is about multiplying, logarithms are about finding the inverse, or the 'how many times do we multiply' part. They can be very useful in simplifying complex calculations or solving exponential equations. To understand what a logarithm means, ask yourself '10 raised to what power gives this number?'
scientific calculator usage
Using a scientific calculator can simplify many mathematical tasks, including finding logarithms. Most scientific calculators have a dedicated log button for base 10 logarithms.
Here's a simple way to use it:
  • Turn on the calculator.
  • Input the number you want to find the logarithm of, in this case, 0.0326.
  • Press the 'log' button.
  • The calculator will display the result.
Depending on your calculator model, the steps might vary slightly. Some models require you to press the log button first and then enter the number. Always refer to your calculator's manual for precise instructions.
decimal approximation
Decimal approximation is essential for making numbers more manageable. When dealing with logarithms, calculators often provide results with many decimal places. In most cases, we need a more concise number. This is where rounding comes in. For example, if the calculator shows a result as -1.48627, we can round it to four decimal places: -1.4863. To do this, we look at the fifth decimal place. If it is 5 or higher, we round the fourth decimal place up. If it is less than 5, we leave the fourth decimal place as is.
Knowing how to round properly ensures accuracy and simplicity in your calculations.

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Most popular questions from this chapter

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$ \log _{10} 2=0.3010 \quad \text { and } \quad \log _{10} 9=0.9542 $$ Evaluate each logarithm by applying the appropriate rule or rules from this section. $$ \log _{10} 2^{19} $$

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$ 3 \log _{a} 5-4 \log _{a} 3 $$

Solve each equation. Use natural logarithms. When appropriate, give solutions to three decimal places. See Example 2. $$ e^{0.012 x}=23 $$

Solve equation. \(\log _{a} 1=0\)

Find the hydronium ion concentration of the substance with the given pH. Human gastric contents, 2.0
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