Simplify The Expression Without Using A Calculator. 10 Log ⁡ 5 X 10^{\log \sqrt{5x}} 1 0 L O G 5 X ​ 10 Log ⁡ 5 X = □ 10^{\log \sqrt{5x}} = \square 1 0 L O G 5 X ​ = □

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Understanding the Problem

The given expression is 10log⁡5x10^{\log \sqrt{5x}}. We are asked to simplify this expression without using a calculator. To simplify this expression, we need to understand the properties of logarithms and exponents.

Properties of Logarithms and Exponents

The expression 10log⁡5x10^{\log \sqrt{5x}} involves both logarithms and exponents. To simplify this expression, we need to use the properties of logarithms and exponents.

Logarithmic Property

One of the fundamental properties of logarithms is that log⁡ab=c\log_a{b} = c is equivalent to ac=ba^c = b. This property can be used to simplify the expression 10log⁡5x10^{\log \sqrt{5x}}.

Exponential Property

Another important property of exponents is that alog⁡ab=ba^{\log_a{b}} = b. This property can also be used to simplify the expression 10log⁡5x10^{\log \sqrt{5x}}.

Simplifying the Expression

Using the properties of logarithms and exponents, we can simplify the expression 10log⁡5x10^{\log \sqrt{5x}} as follows:

Step 1: Apply the Logarithmic Property

The expression 10log⁡5x10^{\log \sqrt{5x}} can be rewritten as 5x\sqrt{5x} using the logarithmic property.

Step 2: Apply the Exponential Property

The expression 5x\sqrt{5x} can be rewritten as 5x5x using the exponential property.

Step 3: Simplify the Expression

The expression 5x5x is already simplified.

Final Answer

The final answer is 5x\boxed{5x}.

Conclusion

In this article, we simplified the expression 10log⁡5x10^{\log \sqrt{5x}} without using a calculator. We used the properties of logarithms and exponents to simplify the expression. The final answer is 5x\boxed{5x}.

Frequently Asked Questions

Q: What is the property of logarithms used in this problem?

A: The logarithmic property used in this problem is log⁡ab=c\log_a{b} = c is equivalent to ac=ba^c = b.

Q: What is the property of exponents used in this problem?

A: The exponential property used in this problem is alog⁡ab=ba^{\log_a{b}} = b.

Q: How do we simplify the expression 10log⁡5x10^{\log \sqrt{5x}}?

A: We simplify the expression 10log⁡5x10^{\log \sqrt{5x}} by applying the logarithmic property and the exponential property.

References

Related Topics

  • [1] Simplifying Expressions with Exponents
  • [2] Simplifying Expressions with Logarithms
  • [3] Properties of Logarithms and Exponents

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Q&A: Simplifying Expressions with Logarithms and Exponents

Q: What is the property of logarithms used in simplifying expressions?

A: The logarithmic property used in simplifying expressions is log⁡ab=c\log_a{b} = c is equivalent to ac=ba^c = b. This property can be used to rewrite an expression in a simpler form.

Q: What is the property of exponents used in simplifying expressions?

A: The exponential property used in simplifying expressions is alog⁡ab=ba^{\log_a{b}} = b. This property can be used to rewrite an expression in a simpler form.

Q: How do we simplify the expression 10log⁡5x10^{\log \sqrt{5x}}?

A: We simplify the expression 10log⁡5x10^{\log \sqrt{5x}} by applying the logarithmic property and the exponential property. First, we rewrite the expression as 5x\sqrt{5x} using the logarithmic property. Then, we rewrite the expression as 5x5x using the exponential property.

Q: What is the final answer to the expression 10log⁡5x10^{\log \sqrt{5x}}?

A: The final answer to the expression 10log⁡5x10^{\log \sqrt{5x}} is 5x\boxed{5x}.

Q: Can we simplify expressions with logarithms and exponents using a calculator?

A: Yes, we can simplify expressions with logarithms and exponents using a calculator. However, in this article, we simplified the expression without using a calculator.

Q: What are some common mistakes to avoid when simplifying expressions with logarithms and exponents?

A: Some common mistakes to avoid when simplifying expressions with logarithms and exponents include:

  • Forgetting to apply the logarithmic property or the exponential property
  • Not rewriting the expression in a simpler form
  • Not checking the final answer for accuracy

Q: How do we check the final answer for accuracy?

A: We check the final answer for accuracy by plugging it back into the original expression and simplifying it. If the final answer is correct, the expression should simplify to the original expression.

Additional FAQs

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

A: A logarithm is the inverse operation of an exponent. In other words, if ab=ca^b = c, then log⁡ac=b\log_a{c} = b.

Q: How do we evaluate an expression with a logarithm?

A: We evaluate an expression with a logarithm by rewriting it in a simpler form using the logarithmic property.

Q: How do we evaluate an expression with an exponent?

A: We evaluate an expression with an exponent by rewriting it in a simpler form using the exponential property.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions with logarithms and exponents. We covered topics such as the properties of logarithms and exponents, how to simplify expressions, and how to check the final answer for accuracy. We hope this article has been helpful in understanding how to simplify expressions with logarithms and exponents.

Related Topics

  • [1] Simplifying Expressions with Exponents
  • [2] Simplifying Expressions with Logarithms
  • [3] Properties of Logarithms and Exponents

References