Applying Log Properties: ExampleOriginal Equation: Log ⁡ 4 5 + 2 Log ⁡ 4 3 − Log ⁡ 4 15 = X \log _4 5 + 2 \log _4 3 - \log _4 15 = X Lo G 4 ​ 5 + 2 Lo G 4 ​ 3 − Lo G 4 ​ 15 = X Step 1: Apply The Power Rule Property: Log ⁡ 4 5 + Log ⁡ 4 3 2 − Log ⁡ 4 15 = X \log _4 5 + \log _4 3^2 - \log _4 15 = X Lo G 4 ​ 5 + Lo G 4 ​ 3 2 − Lo G 4 ​ 15 = X Step 2: Simplify:$\log _4 5 + \log _4 9 - \log _4 15 =

Introduction

Logarithmic equations can be complex and challenging to solve, but with the right properties and techniques, they can be simplified and solved with ease. In this article, we will explore the power rule property of logarithms and apply it to a given equation to simplify it. We will also discuss the importance of logarithmic properties in mathematics and provide examples of how they can be used in real-world applications.

The Power Rule Property of Logarithms

The power rule property of logarithms states that for any positive real numbers a and b, and any real number c:

$$\log_a b^c = c \log_a b$$

This property allows us to simplify logarithmic expressions by bringing the exponent down as a coefficient. In the context of the given equation, we will apply this property to simplify the expression.

Step 1: Apply the Power Rule Property

The original equation is:

$$\log _4 5 + 2 \log _4 3 - \log _4 15 = x$$

To apply the power rule property, we will rewrite the equation as:

$$\log _4 5 + \log _4 3^2 - \log _4 15 = x$$

Using the power rule property, we can rewrite the second term as:

$$\log _4 3^2 = 2 \log _4 3$$

So, the equation becomes:

$$\log _4 5 + 2 \log _4 3 - \log _4 15 = x$$

Step 2: Simplify

Now that we have applied the power rule property, we can simplify the equation further. We can combine the first two terms using the product rule property, which states that:

$$\log_a b + \log_a c = \log_a (b \cdot c)$$

Using this property, we can rewrite the equation as:

$$\log _4 (5 \cdot 3^2) - \log _4 15 = x$$

Simplifying the expression inside the logarithm, we get:

$$\log _4 (5 \cdot 9) - \log _4 15 = x$$

$$\log _4 45 - \log _4 15 = x$$

Discussion

The power rule property of logarithms is a powerful tool for simplifying logarithmic equations. By applying this property, we can rewrite complex expressions in a simpler form, making it easier to solve the equation. In this example, we applied the power rule property to simplify the given equation and arrived at a simpler expression.

Real-World Applications

Logarithmic properties, including the power rule property, have numerous real-world applications in fields such as:

• Finance: Logarithmic properties are used in finance to calculate interest rates, investment returns, and risk analysis.
• Science: Logarithmic properties are used in science to calculate pH levels, concentrations, and rates of chemical reactions.
• Engineering: Logarithmic properties are used in engineering to calculate stress, strain, and vibration frequencies.

Conclusion

In conclusion, the power rule property of logarithms is a fundamental concept in mathematics that allows us to simplify logarithmic equations. By applying this property, we can rewrite complex expressions in a simpler form, making it easier to solve the equation. This property has numerous real-world applications in fields such as finance, science, and engineering.

Additional Examples

Here are some additional examples of how the power rule property can be applied:

• Example 1: $\log _2 3^4 + \log _2 5 - \log _2 7 = x$
• Example 2: $\log _3 2^3 - \log _3 5 + \log _3 7 = x$
• Example 3: $\log _5 2^2 + \log _5 3 - \log _5 7 = x$

These examples demonstrate the power of the power rule property in simplifying logarithmic equations.

Final Thoughts

Introduction

In our previous article, we explored the power rule property of logarithms and applied it to a given equation to simplify it. In this article, we will provide a Q&A guide to help you understand the power rule property and its applications.

Q&A

Q: What is the power rule property of logarithms?

A: The power rule property of logarithms states that for any positive real numbers a and b, and any real number c:

$$\log_a b^c = c \log_a b$$

This property allows us to simplify logarithmic expressions by bringing the exponent down as a coefficient.

Q: How do I apply the power rule property to a logarithmic equation?

A: To apply the power rule property, you need to identify the exponent in the logarithmic expression and bring it down as a coefficient. For example, if you have the equation:

$$\log _4 5 + 2 \log _4 3 - \log _4 15 = x$$

You can apply the power rule property by rewriting the second term as:

$$\log _4 3^2 = 2 \log _4 3$$

So, the equation becomes:

$$\log _4 5 + 2 \log _4 3 - \log _4 15 = x$$

Q: Can I use the power rule property to simplify logarithmic expressions with different bases?

A: Yes, you can use the power rule property to simplify logarithmic expressions with different bases. For example, if you have the equation:

$$\log _2 3^4 + \log _2 5 - \log _2 7 = x$$

You can apply the power rule property by rewriting the first term as:

$$\log _2 3^4 = 4 \log _2 3$$

So, the equation becomes:

$$4 \log _2 3 + \log _2 5 - \log _2 7 = x$$

Q: How do I use the power rule property to simplify logarithmic expressions with negative exponents?

A: To simplify logarithmic expressions with negative exponents, you need to use the property that:

$$\log_a b^{-c} = -c \log_a b$$

For example, if you have the equation:

$$\log _4 5^{-2} + \log _4 3 - \log _4 15 = x$$

You can apply the power rule property by rewriting the first term as:

$$\log _4 5^{-2} = -2 \log _4 5$$

So, the equation becomes:

$$-2 \log _4 5 + \log _4 3 - \log _4 15 = x$$

Q: Can I use the power rule property to simplify logarithmic expressions with fractional exponents?

A: Yes, you can use the power rule property to simplify logarithmic expressions with fractional exponents. For example, if you have the equation:

$$\log _4 5^{\frac{1}{2}} + \log _4 3 - \log _4 15 = x$$

You can apply the power rule property by rewriting the first term as:

$$\log _4 5^{\frac{1}{2}} = \frac{1}{2} \log _4 5$$

So, the equation becomes:

$$\frac{1}{2} \log _4 5 + \log _4 3 - \log _4 15 = x$$

Conclusion

In conclusion, the power rule property of logarithms is a powerful tool for simplifying logarithmic equations. By applying this property, you can rewrite complex expressions in a simpler form, making it easier to solve the equation. We hope that this Q&A guide has provided a clear understanding of the power rule property and its applications.

Additional Resources

For more information on logarithmic properties and their applications, please refer to the following resources:

• Logarithmic Properties: A comprehensive guide to logarithmic properties, including the power rule property.
• Logarithmic Equations: A guide to solving logarithmic equations, including examples and practice problems.
• Logarithmic Functions: A guide to logarithmic functions, including their properties and applications.

We hope that this article has provided a clear understanding of the power rule property and its applications. If you have any further questions or need additional resources, please don't hesitate to contact us.