Problem 43 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \log _{10} 1,000 $$

Step-by-Step Solution

Answer

The value of \(\log_{10} 1,000\) is 3.

Step 1: Understand the Base

First, identify the base of the logarithm. In this expression, it is log base 10, often written as \(\log_{10}\). This is also known as the common logarithm.

Step 2: Set Up the Equation

The expression \(\log_{10} 1,000\) is asking, 'To what power must 10 be raised to get 1,000?' So, set up the equation: \(10^x = 1,000\).

Step 3: Convert the Number to Exponential Form

Write 1,000 as a power of 10. We know that \(1,000 = 10^3\).

Step 4: Relate the Exponents

Since \(10^x = 10^3\), we can equate the exponents, thus \(x = 3\).

Step 5: Solve and Conclude

The solution to the logarithmic expression is \(x = 3\). Therefore, \(\log_{10} 1,000 = 3\).

Key Concepts

Common Logarithm

The common logarithm is frequently used in mathematics because it has a base of 10. This is represented as \(\log_{10}\), though usually it is simply written as \(\log\) with no base shown. For example, \(\log 100\) is understood as \(\log_{10} 100\). The common logarithm is valuable in various fields such as scientific calculations and engineering because it simplifies expressions and computations.

• Common log is one of the most commonly used logarithms, where the base is always 10.
• It helps in converting between exponential and logarithmic forms easily since our number system is also base 10.
• Many calculators provide a common logarithm function for quick computations.

Exponential Form

Working with logarithms often involves converting numbers into their exponential form. This simply means expressing a number as a power of another number.
When we say \(10^x = 1000\), the number 1000 is written in exponential form as \(10^3\). This shows how many times you need to multiply the base by itself to reach the desired number.

• It is used to simplify multiplication and division in higher mathematics and science.
• Helps in visualizing how a number can expand exponentially or shrink in size.
• Provide a clear method to solve for unknown exponents in equations.

Logarithmic Expressions

Logarithmic expressions are mathematical phrases that involve logarithms; for example, \(\log_{10} 1000\). They are used to express logarithms and relate exponential relationships in a simplified form.

• Logarithmic expressions provide a way to express repeated multiplication as a simple operation.
• They convert exponential forms into a manageable expression for computation.
• Understanding these expressions is essential for simplifying complex multiplicative relationships.

Base of a Logarithm

The base of a logarithm is a critical component as it determines the number system used. In \(\log_{b} x\), \(b\) is the base and \(x\) is the argument. For the common logarithm, the base is 10.
Choosing the base affects how you calculate the logarithmic value and align it to a particular context.

• For natural logarithms, the base \(e\) (approximately 2.718) is used, which is significant in continuous growth models.
• The base in logarithm relates closely to the number you are working with and the intended application.
• It's crucial to understand bases to properly interpret and calculate logarithmic expressions.