Write This Logarithmic Expression As An Exponential Expression: ${ \log [\text{CUBE ROOT}(81)] = \left(\frac{2}{3}\right) }$a. { [\text{CUBE ROOT}(81)] = 9^{\left(\frac{2}{3}\right)}$}$b. [$9 = [\text{CUBE

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Understanding Logarithmic and Exponential Expressions

In mathematics, logarithmic and exponential expressions are two fundamental concepts that are closely related. A logarithmic expression is the inverse of an exponential expression. In this article, we will focus on converting logarithmic expressions to exponential expressions, specifically the expression log⁡[CUBE ROOT(81)]=(23)\log [\text{CUBE ROOT}(81)] = \left(\frac{2}{3}\right).

What is a Logarithmic Expression?

A logarithmic expression is an expression that represents the power or exponent to which a base number must be raised to produce a given value. In other words, it is the inverse of an exponential expression. For example, if we have the expression log⁡10100=2\log_{10} 100 = 2, it means that 102=10010^2 = 100. The logarithmic expression log⁡10100\log_{10} 100 represents the power to which the base number 10 must be raised to produce the value 100.

What is an Exponential Expression?

An exponential expression is an expression that represents a value that is raised to a power. For example, the expression 23=82^3 = 8 represents the value 8 as the result of raising 2 to the power of 3.

Converting Logarithmic Expressions to Exponential Expressions

To convert a logarithmic expression to an exponential expression, we need to use the following formula:

log⁡bx=y  ⟺  by=x\log_{b} x = y \iff b^y = x

where bb is the base number, xx is the value, and yy is the exponent.

Converting the Given Logarithmic Expression

Now, let's apply the formula to convert the given logarithmic expression log⁡[CUBE ROOT(81)]=(23)\log [\text{CUBE ROOT}(81)] = \left(\frac{2}{3}\right) to an exponential expression.

The cube root of 81 can be written as 811381^{\frac{1}{3}}. Therefore, we can rewrite the expression as:

log⁡[CUBE ROOT(81)]=log⁡8113=(23)\log [\text{CUBE ROOT}(81)] = \log 81^{\frac{1}{3}} = \left(\frac{2}{3}\right)

Using the formula, we can rewrite the expression as:

8113=9(23)81^{\frac{1}{3}} = 9^{\left(\frac{2}{3}\right)}

Therefore, the exponential expression equivalent to the given logarithmic expression is:

[CUBE ROOT(81)]=9(23)[\text{CUBE ROOT}(81)] = 9^{\left(\frac{2}{3}\right)}

Conclusion

In this article, we have learned how to convert logarithmic expressions to exponential expressions using the formula log⁡bx=y  ⟺  by=x\log_{b} x = y \iff b^y = x. We have applied this formula to convert the given logarithmic expression log⁡[CUBE ROOT(81)]=(23)\log [\text{CUBE ROOT}(81)] = \left(\frac{2}{3}\right) to an exponential expression. The resulting exponential expression is [CUBE ROOT(81)]=9(23)[\text{CUBE ROOT}(81)] = 9^{\left(\frac{2}{3}\right)}.

Common Mistakes to Avoid

When converting logarithmic expressions to exponential expressions, it is essential to avoid common mistakes such as:

  • Incorrect application of the formula: Make sure to apply the formula correctly by using the base number, value, and exponent.
  • Incorrect simplification: Simplify the expression correctly by using the properties of exponents.
  • Incorrect evaluation: Evaluate the expression correctly by using the properties of logarithms and exponents.

Real-World Applications

Converting logarithmic expressions to exponential expressions has numerous real-world applications in various fields such as:

  • Science: Logarithmic and exponential expressions are used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Logarithmic and exponential expressions are used to design and optimize systems, such as electronic circuits and mechanical systems.
  • Finance: Logarithmic and exponential expressions are used to calculate interest rates, investment returns, and other financial metrics.

Practice Problems

To practice converting logarithmic expressions to exponential expressions, try the following problems:

  • log⁡101000=3\log_{10} 1000 = 3
  • log⁡216=4\log_{2} 16 = 4
  • log⁡5125=3\log_{5} 125 = 3

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about logarithmic and exponential expressions.

Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression is an expression that represents the power or exponent to which a base number must be raised to produce a given value. An exponential expression is an expression that represents a value that is raised to a power.

Q: How do I convert a logarithmic expression to an exponential expression?

A: To convert a logarithmic expression to an exponential expression, use the formula log⁡bx=y  ⟺  by=x\log_{b} x = y \iff b^y = x. This formula states that if log⁡bx=y\log_{b} x = y, then by=xb^y = x.

Q: What is the base number in a logarithmic expression?

A: The base number in a logarithmic expression is the number that is raised to a power to produce a given value. For example, in the expression log⁡10100=2\log_{10} 100 = 2, the base number is 10.

Q: What is the exponent in a logarithmic expression?

A: The exponent in a logarithmic expression is the power to which the base number is raised to produce a given value. For example, in the expression log⁡10100=2\log_{10} 100 = 2, the exponent is 2.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, use the properties of logarithms, such as the product rule and the quotient rule. For example, log⁡10(100⋅1000)=log⁡10100+log⁡101000\log_{10} (100 \cdot 1000) = \log_{10} 100 + \log_{10} 1000.

Q: How do I evaluate an exponential expression?

A: To evaluate an exponential expression, use the properties of exponents, such as the product rule and the quotient rule. For example, 23⋅24=23+4=272^3 \cdot 2^4 = 2^{3+4} = 2^7.

Q: What are some common mistakes to avoid when working with logarithmic and exponential expressions?

A: Some common mistakes to avoid when working with logarithmic and exponential expressions include:

  • Incorrect application of the formula: Make sure to apply the formula correctly by using the base number, value, and exponent.
  • Incorrect simplification: Simplify the expression correctly by using the properties of exponents.
  • Incorrect evaluation: Evaluate the expression correctly by using the properties of logarithms and exponents.

Q: What are some real-world applications of logarithmic and exponential expressions?

A: Logarithmic and exponential expressions have numerous real-world applications in various fields such as:

  • Science: Logarithmic and exponential expressions are used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Logarithmic and exponential expressions are used to design and optimize systems, such as electronic circuits and mechanical systems.
  • Finance: Logarithmic and exponential expressions are used to calculate interest rates, investment returns, and other financial metrics.

Q: How can I practice converting logarithmic expressions to exponential expressions?

A: To practice converting logarithmic expressions to exponential expressions, try the following problems:

  • log⁡101000=3\log_{10} 1000 = 3
  • log⁡216=4\log_{2} 16 = 4
  • log⁡5125=3\log_{5} 125 = 3

Conclusion

In conclusion, logarithmic and exponential expressions are fundamental concepts in mathematics that have numerous real-world applications. By understanding the formula log⁡bx=y  ⟺  by=x\log_{b} x = y \iff b^y = x and applying it correctly, we can convert logarithmic expressions to exponential expressions. Remember to avoid common mistakes and practice converting logarithmic expressions to exponential expressions to become proficient in this skill.