Which Identity Is True?A. Log ⁡ 3.1 85 = Ln ⁡ 8.5 Log ⁡ 3.1 \log _{3.1} 85 = \frac{\ln 8.5}{\log 3.1} Lo G 3.1 ​ 85 = L O G 3.1 L N 8.5 ​

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Introduction

In mathematics, identities are equations that are true for all possible values of the variables involved. They are often used to simplify complex expressions and solve problems. In this article, we will examine two given identities involving logarithms and determine which one is true.

Logarithmic Identities

Logarithmic identities are equations that involve logarithms and are true for all possible values of the variables involved. They are often used to simplify complex expressions and solve problems. There are several types of logarithmic identities, including:

  • Change of Base Formula: This formula allows us to change the base of a logarithm from one base to another. It is given by: log⁡ba=log⁡calog⁡cb\log_b a = \frac{\log_c a}{\log_c b}, where aa, bb, and cc are positive real numbers and c≠1c \neq 1.
  • Product Rule: This rule states that the logarithm of a product is equal to the sum of the logarithms of the factors. It is given by: log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b y, where xx and yy are positive real numbers.
  • Quotient Rule: This rule states that the logarithm of a quotient is equal to the difference of the logarithms of the dividend and the divisor. It is given by: log⁡b(xy)=log⁡bx−log⁡by\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y, where xx and yy are positive real numbers.

Examining the Given Identities

We are given two identities involving logarithms:

A. log⁡3.185=ln⁡8.5log⁡3.1\log _{3.1} 85 = \frac{\ln 8.5}{\log 3.1} B. log⁡3.185=log⁡8.5log⁡3.1\log _{3.1} 85 = \frac{\log 8.5}{\log 3.1}

To determine which identity is true, we need to examine each one separately.

Identity A

Identity A states that log⁡3.185=ln⁡8.5log⁡3.1\log _{3.1} 85 = \frac{\ln 8.5}{\log 3.1}. To determine if this identity is true, we need to apply the change of base formula. The change of base formula states that log⁡ba=log⁡calog⁡cb\log_b a = \frac{\log_c a}{\log_c b}, where aa, bb, and cc are positive real numbers and c≠1c \neq 1. In this case, we can let a=85a = 85, b=3.1b = 3.1, and c=ec = e (the base of the natural logarithm). Then, we have:

log⁡3.185=ln⁡85ln⁡3.1\log _{3.1} 85 = \frac{\ln 85}{\ln 3.1}

This expression is equal to ln⁡8.5log⁡3.1\frac{\ln 8.5}{\log 3.1}, which is the right-hand side of identity A. Therefore, identity A is true.

Identity B

Identity B states that log⁡3.185=log⁡8.5log⁡3.1\log _{3.1} 85 = \frac{\log 8.5}{\log 3.1}. To determine if this identity is true, we need to apply the change of base formula. However, in this case, we cannot let a=85a = 85, b=3.1b = 3.1, and c=ec = e (the base of the natural logarithm), because the change of base formula requires that c≠1c \neq 1. Therefore, we cannot apply the change of base formula to identity B.

However, we can still examine identity B by using the quotient rule. The quotient rule states that log⁡b(xy)=log⁡bx−log⁡by\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y, where xx and yy are positive real numbers. In this case, we can let x=85x = 85 and y=3.1y = 3.1. Then, we have:

log⁡3.185=log⁡3.1(85⋅1)=log⁡3.185−log⁡3.11\log _{3.1} 85 = \log _{3.1} (85 \cdot 1) = \log _{3.1} 85 - \log _{3.1} 1

This expression is equal to log⁡3.185−0\log _{3.1} 85 - 0, which is not equal to log⁡8.5log⁡3.1\frac{\log 8.5}{\log 3.1}. Therefore, identity B is not true.

Conclusion

In conclusion, we have examined two given identities involving logarithms and determined which one is true. Identity A is true, while identity B is not true. The change of base formula and the quotient rule were used to examine each identity separately.

Final Thoughts

Logarithmic identities are an important part of mathematics, and they are used to simplify complex expressions and solve problems. In this article, we have examined two given identities involving logarithms and determined which one is true. The change of base formula and the quotient rule were used to examine each identity separately. We hope that this article has provided a clear understanding of logarithmic identities and how they can be used to solve problems.

References

Introduction

In our previous article, we examined two given identities involving logarithms and determined which one is true. In this article, we will provide a Q&A section to help clarify any questions or doubts that readers may have.

Q&A

Q: What is the change of base formula?

A: The change of base formula is a logarithmic identity that allows us to change the base of a logarithm from one base to another. It is given by: log⁡ba=log⁡calog⁡cb\log_b a = \frac{\log_c a}{\log_c b}, where aa, bb, and cc are positive real numbers and c≠1c \neq 1.

Q: What is the quotient rule?

A: The quotient rule is a logarithmic identity that states that the logarithm of a quotient is equal to the difference of the logarithms of the dividend and the divisor. It is given by: log⁡b(xy)=log⁡bx−log⁡by\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y, where xx and yy are positive real numbers.

Q: How do I apply the change of base formula?

A: To apply the change of base formula, you need to identify the base of the logarithm that you want to change, and then use the formula: log⁡ba=log⁡calog⁡cb\log_b a = \frac{\log_c a}{\log_c b}, where aa, bb, and cc are positive real numbers and c≠1c \neq 1.

Q: Can I use the change of base formula with any base?

A: No, you cannot use the change of base formula with any base. The base cc must be a positive real number and must not be equal to 1.

Q: What is the difference between the change of base formula and the quotient rule?

A: The change of base formula allows us to change the base of a logarithm from one base to another, while the quotient rule allows us to simplify the logarithm of a quotient.

Q: Can I use the quotient rule with any dividend and divisor?

A: No, you cannot use the quotient rule with any dividend and divisor. The dividend and divisor must be positive real numbers.

Q: How do I simplify a logarithmic expression using the quotient rule?

A: To simplify a logarithmic expression using the quotient rule, you need to identify the dividend and divisor, and then use the formula: log⁡b(xy)=log⁡bx−log⁡by\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y, where xx and yy are positive real numbers.

Q: Can I use the change of base formula and the quotient rule together?

A: Yes, you can use the change of base formula and the quotient rule together to simplify a logarithmic expression.

Conclusion

In conclusion, we have provided a Q&A section to help clarify any questions or doubts that readers may have about logarithmic identities. We hope that this article has provided a clear understanding of logarithmic identities and how they can be used to simplify complex expressions and solve problems.

Final Thoughts

Logarithmic identities are an important part of mathematics, and they are used to simplify complex expressions and solve problems. In this article, we have provided a Q&A section to help clarify any questions or doubts that readers may have about logarithmic identities. We hope that this article has provided a clear understanding of logarithmic identities and how they can be used to solve problems.

References