Evaluate The Expression: $\log_{10} \sqrt{35} - \log_{10} \sqrt{7} + \log_{10} \sqrt{2}$

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Introduction

In this article, we will evaluate the given expression $\log_{10} \sqrt{35} - \log_{10} \sqrt{7} + \log_{10} \sqrt{2}$ using the properties of logarithms. We will start by simplifying the expression using the properties of logarithms and then evaluate the resulting expression.

Properties of Logarithms

Before we start evaluating the expression, let's recall some of the properties of logarithms that we will use in this article.

• Product Rule: $\log_{a} (xy) = \log_{a} x + \log_{a} y$
• Quotient Rule: $\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$
• Power Rule: $\log_{a} x^{n} = n \log_{a} x$

Simplifying the Expression

Now that we have recalled the properties of logarithms, let's simplify the given expression.

$\log_{10} \sqrt{35} - \log_{10} \sqrt{7} + \log_{10} \sqrt{2}$

We can start by simplifying each term separately using the properties of logarithms.

$\log_{10} \sqrt{35} = \log_{10} (35^{\frac{1}{2}}) = \frac{1}{2} \log_{10} 35$

$\log_{10} \sqrt{7} = \log_{10} (7^{\frac{1}{2}}) = \frac{1}{2} \log_{10} 7$

$\log_{10} \sqrt{2} = \log_{10} (2^{\frac{1}{2}}) = \frac{1}{2} \log_{10} 2$

Now that we have simplified each term, let's substitute them back into the original expression.

$\frac{1}{2} \log_{10} 35 - \frac{1}{2} \log_{10} 7 + \frac{1}{2} \log_{10} 2$

Using the Quotient Rule

We can simplify the expression further by using the quotient rule.

$\frac{1}{2} \log_{10} 35 - \frac{1}{2} \log_{10} 7 + \frac{1}{2} \log_{10} 2 = \frac{1}{2} (\log_{10} 35 - \log_{10} 7 + \log_{10} 2)$

Now that we have used the quotient rule, let's simplify the expression inside the parentheses.

$\log_{10} 35 - \log_{10} 7 + \log_{10} 2 = \log_{10} \frac{35}{7} + \log_{10} 2$

Using the Product Rule

We can simplify the expression further by using the product rule.

$\log_{10} \frac{35}{7} + \log_{10} 2 = \log_{10} \frac{35}{7} \cdot 2 = \log_{10} \frac{70}{7}$

Evaluating the Expression

Now that we have simplified the expression, let's evaluate it.

$\log_{10} \frac{70}{7} = \log_{10} 10 = 1$

Conclusion

In this article, we evaluated the given expression $\log_{10} \sqrt{35} - \log_{10} \sqrt{7} + \log_{10} \sqrt{2}$ using the properties of logarithms. We started by simplifying the expression using the properties of logarithms and then evaluated the resulting expression. The final answer is $\boxed{1}$.

Final Answer

The final answer is $\boxed{1}$.

Related Topics

• Logarithmic Equations: Logarithmic equations are equations that contain logarithmic expressions. They can be solved using the properties of logarithms.
• Exponential Equations: Exponential equations are equations that contain exponential expressions. They can be solved using the properties of exponents.
• Trigonometric Equations: Trigonometric equations are equations that contain trigonometric expressions. They can be solved using the properties of trigonometric functions.

References

• "Logarithms" by Math Open Reference. Math Open Reference is a free online mathematics reference book that provides detailed explanations and examples of various mathematical concepts.
• "Exponents and Logarithms" by Khan Academy. Khan Academy is a free online education platform that provides detailed explanations and examples of various mathematical concepts.
• "Trigonometry" by Wolfram MathWorld. Wolfram MathWorld is a free online mathematics reference book that provides detailed explanations and examples of various mathematical concepts.
  

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Introduction

In our previous article, we evaluated the expression $\log_{10} \sqrt{35} - \log_{10} \sqrt{7} + \log_{10} \sqrt{2}$ using the properties of logarithms. In this article, we will answer some of the frequently asked questions related to this topic.

Q1: What is the property of logarithms that is used to simplify the expression $\log_{10} \sqrt{35} - \log_{10} \sqrt{7} + \log_{10} \sqrt{2}$?

A1: The property of logarithms that is used to simplify the expression is the power rule, which states that $\log_{a} x^{n} = n \log_{a} x$.

Q2: How do you simplify the expression $\log_{10} \sqrt{35} - \log_{10} \sqrt{7} + \log_{10} \sqrt{2}$ using the power rule?

A2: To simplify the expression using the power rule, we can rewrite each term as follows:

$\log_{10} \sqrt{35} = \log_{10} (35^{\frac{1}{2}}) = \frac{1}{2} \log_{10} 35$

$\log_{10} \sqrt{7} = \log_{10} (7^{\frac{1}{2}}) = \frac{1}{2} \log_{10} 7$

$\log_{10} \sqrt{2} = \log_{10} (2^{\frac{1}{2}}) = \frac{1}{2} \log_{10} 2$

Q3: What is the next step in simplifying the expression $\log_{10} \sqrt{35} - \log_{10} \sqrt{7} + \log_{10} \sqrt{2}$?

A3: The next step in simplifying the expression is to substitute the simplified terms back into the original expression.

$\frac{1}{2} \log_{10} 35 - \frac{1}{2} \log_{10} 7 + \frac{1}{2} \log_{10} 2$

Q4: How do you simplify the expression $\frac{1}{2} \log_{10} 35 - \frac{1}{2} \log_{10} 7 + \frac{1}{2} \log_{10} 2$ further?

A4: To simplify the expression further, we can use the quotient rule, which states that $\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$.

$\frac{1}{2} \log_{10} 35 - \frac{1}{2} \log_{10} 7 + \frac{1}{2} \log_{10} 2 = \frac{1}{2} (\log_{10} 35 - \log_{10} 7 + \log_{10} 2)$

Q5: What is the final step in simplifying the expression $\log_{10} \sqrt{35} - \log_{10} \sqrt{7} + \log_{10} \sqrt{2}$?

A5: The final step in simplifying the expression is to evaluate the expression inside the parentheses.

$\log_{10} 35 - \log_{10} 7 + \log_{10} 2 = \log_{10} \frac{35}{7} + \log_{10} 2$

Q6: How do you simplify the expression $\log_{10} \frac{35}{7} + \log_{10} 2$ further?

A6: To simplify the expression further, we can use the product rule, which states that $\log_{a} (xy) = \log_{a} x + \log_{a} y$.

$\log_{10} \frac{35}{7} + \log_{10} 2 = \log_{10} \frac{35}{7} \cdot 2 = \log_{10} \frac{70}{7}$

Q7: What is the final answer to the expression $\log_{10} \sqrt{35} - \log_{10} \sqrt{7} + \log_{10} \sqrt{2}$?

A7: The final answer to the expression is $\boxed{1}$.

Conclusion

In this article, we answered some of the frequently asked questions related to the expression $\log_{10} \sqrt{35} - \log_{10} \sqrt{7} + \log_{10} \sqrt{2}$. We hope that this article has been helpful in understanding the properties of logarithms and how to simplify expressions using these properties.

Final Answer

The final answer is $\boxed{1}$.

Related Topics

• Logarithmic Equations: Logarithmic equations are equations that contain logarithmic expressions. They can be solved using the properties of logarithms.
• Exponential Equations: Exponential equations are equations that contain exponential expressions. They can be solved using the properties of exponents.
• Trigonometric Equations: Trigonometric equations are equations that contain trigonometric expressions. They can be solved using the properties of trigonometric functions.

References

• "Logarithms" by Math Open Reference. Math Open Reference is a free online mathematics reference book that provides detailed explanations and examples of various mathematical concepts.
• "Exponents and Logarithms" by Khan Academy. Khan Academy is a free online education platform that provides detailed explanations and examples of various mathematical concepts.
• "Trigonometry" by Wolfram MathWorld. Wolfram MathWorld is a free online mathematics reference book that provides detailed explanations and examples of various mathematical concepts.