The Given Logarithmic Values

Oct 13, 2025 by miodeya.com 5 views

Introduction

In mathematics, logarithms play a crucial role in solving various problems, especially those involving exponential functions. Given a set of logarithmic values, we can use various techniques to find the value of a logarithm that is not directly provided. In this article, we will explore how to find the value of $\log 47$ using the given logarithmic values.

The Given Logarithmic Values

We are given the following logarithmic values:

\log 2 = 0.3010, \quad \log 3 = 0.4771, \quad \log 5 = 0.6990, \quad \log 7 = 0.8451

Step 1: Expressing 47 as a Product of Prime Factors

To find the value of $\log 47$ , we need to express 47 as a product of prime factors. Since 47 is a prime number, it can only be expressed as a product of 1 and itself.

47 = 1 \times 47

Step 2: Using the Logarithmic Product Rule

The logarithmic product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. We can use this rule to express $\log 47$ in terms of the given logarithmic values.

\log 47 = \log (1 \times 47) = \log 1 + \log 47

However, we know that $\log 1 = 0$ , so we can simplify the expression as follows:

\log 47 = 0 + \log 47 = \log 47

Step 3: Expressing 47 as a Product of Prime Factors Using the Given Logarithmic Values

We can express 47 as a product of prime factors using the given logarithmic values. Since 47 is a prime number, we can write it as a product of 2, 3, 5, and 7.

47 = 2 \times 3 \times 5 \times 7

Step 4: Using the Logarithmic Product Rule Again

We can use the logarithmic product rule again to express $\log 47$ in terms of the given logarithmic values.

\log 47 = \log (2 \times 3 \times 5 \times 7) = \log 2 + \log 3 + \log 5 + \log 7

Step 5: Substituting the Given Logarithmic Values

We can substitute the given logarithmic values into the expression.

\log 47 = 0.3010 + 0.4771 + 0.6990 + 0.8451

Step 6: Evaluating the Expression

We can evaluate the expression to find the value of $\log 47$ .

\log 47 = 0.3010 + 0.4771 + 0.6990 + 0.8451 = 2.2222

Conclusion

In this article, we used various techniques to find the value of $\log 47$ using the given logarithmic values. We expressed 47 as a product of prime factors, used the logarithmic product rule, and substituted the given logarithmic values to find the value of $\log 47$ . The final answer is $\boxed{2.2222}$ .

Discussion

The value of $\log 47$ can be used to solve various problems involving exponential functions. For example, we can use the value of $\log 47$ to find the value of $47^x$ for a given value of $x$ . We can also use the value of $\log 47$ to find the value of $\frac{47^x}{47^y}$ for given values of $x$ and $y$ .

References

[1] "Logarithms" by Math Open Reference
[2] "Logarithmic Product Rule" by Khan Academy

Additional Resources

[1] "Logarithms" by Wolfram MathWorld
[2] "Logarithmic Product Rule" by MIT OpenCourseWare
Logarithm Q&A: Frequently Asked Questions =============================================

Introduction

In our previous article, we explored how to find the value of $\log 47$ using the given logarithmic values. In this article, we will answer some frequently asked questions about logarithms and provide additional resources for further learning.

Q: What is a logarithm?

A: A logarithm is the inverse operation of exponentiation. It is a mathematical function that takes a number and returns the power to which a base number must be raised to produce that number.

Q: What is the difference between a logarithm and an exponent?

A: A logarithm and an exponent are inverse operations. A logarithm takes a number and returns the power to which a base number must be raised to produce that number, while an exponent takes a base number and returns the result of raising that base number to a given power.

Q: What are the common logarithmic bases?

A: The most common logarithmic bases are 10 and e (approximately 2.71828). Logarithms with base 10 are called common logarithms, while logarithms with base e are called natural logarithms.

Q: How do I calculate a logarithm?

A: To calculate a logarithm, you can use a calculator or a logarithmic table. Alternatively, you can use the logarithmic product rule and the logarithmic power rule to express the logarithm in terms of simpler logarithms.

Q: What is the logarithmic product rule?

A: The logarithmic product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this can be expressed as:

\log (ab) = \log a + \log b

Q: What is the logarithmic power rule?

A: The logarithmic power rule states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. Mathematically, this can be expressed as:

\log a^b = b \log a

Q: How do I use a logarithmic table?

A: A logarithmic table is a table that lists the logarithms of various numbers. To use a logarithmic table, you can look up the logarithm of a number in the table and use it to calculate the logarithm of a product or a power.

Q: What are some common logarithmic identities?

A: Some common logarithmic identities include:

$\log 1 = 0$
$\log a^b = b \log a$
$\log (ab) = \log a + \log b$
$\log \frac{a}{b} = \log a - \log b$

Q: How do I apply logarithmic identities to solve problems?

A: To apply logarithmic identities to solve problems, you can use the following steps:

Identify the logarithmic identity that is relevant to the problem.
Substitute the values into the identity.
Simplify the expression.
Use the result to solve the problem.

Conclusion

In this article, we answered some frequently asked questions about logarithms and provided additional resources for further learning. We hope that this article has been helpful in understanding logarithms and how to apply them to solve problems.

Discussion

Logarithms are a fundamental concept in mathematics and are used in a wide range of applications, including science, engineering, and finance. By understanding logarithms and how to apply them, you can solve a wide range of problems and make informed decisions.

References

[1] "Logarithms" by Math Open Reference
[2] "Logarithmic Product Rule" by Khan Academy
[3] "Logarithmic Power Rule" by MIT OpenCourseWare

Additional Resources

[1] "Logarithms" by Wolfram MathWorld
[2] "Logarithmic Identities" by Mathway
[3] "Logarithmic Tables" by Calculator Soup