(a) Log ⁡ ( 50 ) 1 / 3 ( 30 ) 2 / 3 \log \frac{(50)^{1 / 3}}{(30)^{2 / 3}} Lo G ( 30 ) 2/3 ( 50 ) 1/3 ​

Introduction

In mathematics, logarithms are a powerful tool for simplifying complex expressions and solving equations. One of the key properties of logarithms is the ability to manipulate expressions involving exponents and fractions. In this article, we will explore how to simplify a complex logarithmic expression using the properties of logarithms.

The Problem

The problem we will be working on is:

$$\log \frac{(50)^{1 / 3}}{(30)^{2 / 3}}$$

This expression involves a fraction with two terms, each raised to a power. Our goal is to simplify this expression using the properties of logarithms.

Step 1: Apply the Quotient Rule

The first step in simplifying this expression is to apply the quotient rule, which states that:

$$\log \frac{a}{b} = \log a - \log b$$

Using this rule, we can rewrite the expression as:

$$\log \frac{(50)^{1 / 3}}{(30)^{2 / 3}} = \log (50)^{1 / 3} - \log (30)^{2 / 3}$$

Step 2: Apply the Power Rule

The next step is to apply the power rule, which states that:

$$\log a^b = b \log a$$

Using this rule, we can rewrite the expression as:

$$\log (50)^{1 / 3} - \log (30)^{2 / 3} = \frac{1}{3} \log 50 - \frac{2}{3} \log 30$$

Step 3: Simplify the Expression

Now that we have applied the quotient and power rules, we can simplify the expression further. To do this, we can use the fact that:

$$\log a - \log b = \log \frac{a}{b}$$

Using this fact, we can rewrite the expression as:

$$\frac{1}{3} \log 50 - \frac{2}{3} \log 30 = \log \left(\frac{50}{30}\right)^{1/3}$$

Step 4: Simplify the Fraction

The final step is to simplify the fraction inside the logarithm. To do this, we can use the fact that:

$$\frac{a}{b} = \frac{a}{b}$$

Using this fact, we can rewrite the expression as:

$$\log \left(\frac{50}{30}\right)^{1/3} = \log \left(\frac{5}{3}\right)^{1/3}$$

Conclusion

In this article, we have seen how to simplify a complex logarithmic expression using the properties of logarithms. We started by applying the quotient rule, followed by the power rule, and finally simplified the expression further using the fact that:

$$\log a - \log b = \log \frac{a}{b}$$

The final expression we obtained was:

$$\log \left(\frac{5}{3}\right)^{1/3}$$

This expression is much simpler than the original expression, and it can be evaluated using a calculator or a logarithmic table.

Key Takeaways

• The quotient rule states that: $\log \frac{a}{b} = \log a - \log b$
• The power rule states that: $\log a^b = b \log a$
• The fact that: $\log a - \log b = \log \frac{a}{b}$ can be used to simplify expressions involving logarithms.

Real-World Applications

Logarithmic expressions are used in a wide range of real-world applications, including:

• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithmic expressions are used to calculate the power and voltage of electrical circuits.

Conclusion

In conclusion, logarithmic expressions are a powerful tool for simplifying complex expressions and solving equations. By applying the quotient and power rules, and using the fact that:

$$\log a - \log b = \log \frac{a}{b}$$

Introduction

In our previous article, we explored how to simplify complex logarithmic expressions using the properties of logarithms. In this article, we will answer some of the most frequently asked questions about logarithmic properties and simplifying complex expressions.

Q: What is the quotient rule for logarithms?

A: The quotient rule for logarithms states that:

$$\log \frac{a}{b} = \log a - \log b$$

This rule allows us to simplify expressions involving fractions by breaking them down into separate logarithmic terms.

Q: What is the power rule for logarithms?

A: The power rule for logarithms states that:

$$\log a^b = b \log a$$

This rule allows us to simplify expressions involving exponents by breaking them down into separate logarithmic terms.

Q: How do I simplify a complex logarithmic expression?

A: To simplify a complex logarithmic expression, you can follow these steps:

1. Apply the quotient rule to break down the expression into separate logarithmic terms.
2. Apply the power rule to simplify the exponents.
3. Use the fact that:

$$\log a - \log b = \log \frac{a}{b}$$

to simplify the expression further.

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

A: A logarithmic expression is an expression that involves a logarithm, such as:

$$\log a$$

An exponential expression is an expression that involves an exponent, such as:

$$a^b$$

While logarithmic and exponential expressions are related, they are not the same thing.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use a calculator or a logarithmic table. Alternatively, you can use the fact that:

$$\log a = \frac{\ln a}{\ln 10}$$

to evaluate the expression using a calculator or a computer.

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

A: Logarithmic expressions have many real-world applications, including:

• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithmic expressions are used to calculate the power and voltage of electrical circuits.

Q: Can I use logarithmic expressions to solve equations?

A: Yes, you can use logarithmic expressions to solve equations. By applying the properties of logarithms, you can simplify complex equations and solve for the unknown variable.

Conclusion

In conclusion, logarithmic expressions are a powerful tool for simplifying complex expressions and solving equations. By applying the quotient and power rules, and using the fact that:

$$\log a - \log b = \log \frac{a}{b}$$

we can simplify complex logarithmic expressions and obtain a much simpler expression that can be evaluated using a calculator or a logarithmic table. We hope that this article has been helpful in answering some of the most frequently asked questions about logarithmic properties and simplifying complex expressions.

Key Takeaways

• The quotient rule states that: $\log \frac{a}{b} = \log a - \log b$
• The power rule states that: $\log a^b = b \log a$
• The fact that: $\log a - \log b = \log \frac{a}{b}$ can be used to simplify expressions involving logarithms.
• Logarithmic expressions have many real-world applications, including finance, science, and engineering.

Real-World Examples

• Finance: A bank uses logarithmic expressions to calculate the interest rate on a loan.
• Science: A scientist uses logarithmic expressions to calculate the pH of a solution.
• Engineering: An engineer uses logarithmic expressions to calculate the power and voltage of an electrical circuit.

Conclusion

In conclusion, logarithmic expressions are a powerful tool for simplifying complex expressions and solving equations. By applying the quotient and power rules, and using the fact that:

$$\log a - \log b = \log \frac{a}{b}$$

we can simplify complex logarithmic expressions and obtain a much simpler expression that can be evaluated using a calculator or a logarithmic table. We hope that this article has been helpful in answering some of the most frequently asked questions about logarithmic properties and simplifying complex expressions.