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Chapter 13: Problem 6

Find the common logarithm of each of the given numbers by using a calculator. $$3.19^{3}$$

Short Answer

Expert verified
The common logarithm of \(3.19^3\) is approximately 1.512.

Step by step solution

01

Calculate the Power

First, we need to compute the value of the expression \(3.19^3\). This number represents 3.19 multiplied by itself three times.
02

Evaluate the Expression

Use your calculator to find \(3.19^3\). Enter 3.19, then use the power function (often labelled as `x^y` or `^`) and raise it to the power of 3. The result of \(3.19^3\) is approximately 32.536159.
03

Find the Common Logarithm

Now, use your calculator to find the common (base 10) logarithm of the result. Enter 32.536159 and use the logarithm function, usually labelled as `log`. The common logarithm of 32.536159 is approximately 1.512.

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of another number, known as the exponent. It is a way to express repeated multiplication. For example, in the expression \(3.19^3\), \(3.19\) is the base, and \(3\) is the exponent, meaning \(3.19\) is multiplied by itself twice more (i.e., \(3.19 \times 3.19 \times 3.19\)).
When dealing with exponentiation, it's important to remember:
  • The base number can be any real number, including fractions and decimals.
  • The exponent indicates how many times the base is used as a factor.
  • An exponent of 2 is often called squaring, and an exponent of 3 is called cubing.
Exponentiation is an essential concept foundational to algebra and also pivotal in understanding logarithmic functions since logarithms are essentially the inverse of exponentiation.
Calculator Usage
Calculators are indispensable tools when working with mathematical expressions, especially when dealing with complex operations like exponentiation and logarithms.
To calculate expressions like \(3.19^3\), you would typically use the power function on your calculator, which may be marked as `x^y` or simply `^`. Here are the steps:
  • Enter the base number: 3.19.
  • Press the `x^y` button.
  • Enter the exponent: 3.
  • Press the equal sign to get the result, which is approximately 32.536159.
Next, to find the common logarithm using a calculator, follow these steps:
  • Enter the number you want the logarithm of: 32.536159.
  • Press the `log` function.
  • The result should appear as approximately 1.512.
Modern calculators often have dedicated buttons for both powers and logarithms, making it straightforward to handle these mathematical operations.
Logarithmic Functions
Logarithmic functions are mathematical expressions that represent the inverse of exponentiation.The common logarithm specifically deals with base 10, symbolized as \(\log_{10}\) or just `log`. This makes them especially useful when calculating growth processes or scientific data, which often scale exponentially.
In the case of the exercise, you used the common logarithm to find \(\log_{10}(32.536159)\), resulting in approximately 1.512. This tells us that 10 raised to the power of 1.512 will approximate 32.536159.
  • Logarithms translate multiplicative processes into additive ones, simplifying complex calculations.
  • The common logarithm is used ubiquitously in science and engineering due to its base matching the decimal system.
  • Understanding logarithms can also help simplify solving many types of equations.
Thereby, logarithms not only assist in calculations but also deepen understanding of exponential relationships and growth patterns in data and natural phenomena.

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Most popular questions from this chapter

Use a calculator to solve the given equations. According to one model, the number \(N\) of Americans (in millions) age 65 and older that will have Alzheimer's disease \(t\) years after 2015 is given by \(N=5.1(1.03)^{t} .\) In what year will this number reach 8.0 million?

Solve the given equations. $$2 \log _{2} 3-\log _{2} x=\log _{2} 45$$

Determine the exact value of each of the given expressions. $$\log _{2}\left(\frac{1}{32}\right)$$

Express each as a sum, difference, or multiple of logarithms. See Example 2. $$10 \log _{5} \sqrt{t}$$

Determine the value of the unknown. $$\log _{7} y=3$$
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