Simplify The Expression: $\log_2\left((u^5 \cdot V)^2\right$\]

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Introduction

In this article, we will simplify the given expression $\log_2\left((u^5 \cdot v)^2\right)$. This involves applying the properties of logarithms and exponents to simplify the expression. We will use the properties of logarithms, such as the power rule and the product rule, to simplify the expression.

Understanding the Properties of Logarithms

Before we simplify the expression, let's review the properties of logarithms. The power rule of logarithms states that $\log_a(b^c) = c \cdot \log_a(b)$. This means that we can bring the exponent down as a coefficient. The product rule of logarithms states that $\log_a(b \cdot c) = \log_a(b) + \log_a(c)$. This means that we can combine the logarithms of two numbers.

Simplifying the Expression

Now that we have reviewed the properties of logarithms, let's simplify the expression $\log_2\left((u^5 \cdot v)^2\right)$. We can start by applying the power rule of logarithms. This states that $\log_a(b^c) = c \cdot \log_a(b)$. In this case, we have $(u^5 \cdot v)^2$, which can be written as $(u^5 \cdot v) \cdot (u^5 \cdot v)$. We can now apply the power rule of logarithms to simplify the expression.

$\log_2\left((u^5 \cdot v)^2\right) = \log_2\left((u^5 \cdot v) \cdot (u^5 \cdot v)\right)$

Using the product rule of logarithms, we can combine the logarithms of the two numbers.

$\log_2\left((u^5 \cdot v) \cdot (u^5 \cdot v)\right) = \log_2(u^5 \cdot v) + \log_2(u^5 \cdot v)$

Now, we can apply the power rule of logarithms to simplify the expression.

$\log_2(u^5 \cdot v) + \log_2(u^5 \cdot v) = 2 \cdot \log_2(u^5 \cdot v)$

Using the product rule of logarithms, we can combine the logarithms of the two numbers.

$2 \cdot \log_2(u^5 \cdot v) = 2 \cdot (\log_2(u^5) + \log_2(v))$

Now, we can apply the power rule of logarithms to simplify the expression.

$2 \cdot (\log_2(u^5) + \log_2(v)) = 2 \cdot (5 \cdot \log_2(u) + \log_2(v))$

Simplifying further, we get:

$2 \cdot (5 \cdot \log_2(u) + \log_2(v)) = 10 \cdot \log_2(u) + 2 \cdot \log_2(v)$

Conclusion

In this article, we simplified the expression $\log_2\left((u^5 \cdot v)^2\right)$. We applied the properties of logarithms, such as the power rule and the product rule, to simplify the expression. We started by applying the power rule of logarithms, which states that $\log_a(b^c) = c \cdot \log_a(b)$. We then applied the product rule of logarithms, which states that $\log_a(b \cdot c) = \log_a(b) + \log_a(c)$. Finally, we simplified the expression further by applying the power rule of logarithms again.

Final Answer

The final answer is: $\boxed{10 \cdot \log_2(u) + 2 \cdot \log_2(v)}$

References

• [1] "Logarithms" by Khan Academy
• [2] "Properties of Logarithms" by Math Is Fun
• [3] "Simplifying Logarithmic Expressions" by Purplemath

Related Topics

• Simplifying Exponential Expressions
• Simplifying Trigonometric Expressions
• Simplifying Algebraic Expressions

Frequently Asked Questions

• Q: What is the power rule of logarithms? A: The power rule of logarithms states that $\log_a(b^c) = c \cdot \log_a(b)$.
• Q: What is the product rule of logarithms? A: The product rule of logarithms states that $\log_a(b \cdot c) = \log_a(b) + \log_a(c)$.
• Q: How do I simplify a logarithmic expression? A: To simplify a logarithmic expression, you can apply the power rule and the product rule of logarithms.
  Frequently Asked Questions: Simplifying Logarithmic Expressions =============================================================

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log_a(b^c) = c \cdot \log_a(b)$. This means that we can bring the exponent down as a coefficient.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log_a(b \cdot c) = \log_a(b) + \log_a(c)$. This means that we can combine the logarithms of two numbers.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can apply the power rule and the product rule of logarithms. Here's a step-by-step guide:

1. Identify the base and the argument of the logarithm.
2. Apply the power rule of logarithms to bring the exponent down as a coefficient.
3. Apply the product rule of logarithms to combine the logarithms of two numbers.
4. Simplify the expression further by combining like terms.

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 $\log_a(b)$. An exponential expression is an expression that involves an exponent, such as $a^b$. While both types of expressions involve powers, they are used to represent different mathematical relationships.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to find the value of the argument of the logarithm. For example, if we have the expression $\log_2(16)$, we need to find the value of $16$ in terms of the base $2$. In this case, $16 = 2^4$, so the value of the expression is $4$.

Q: What is the relationship between logarithms and exponents?

A: Logarithms and exponents are related in that they are inverse operations. This means that if we have an exponential expression, we can take the logarithm of it to get back to the original value. For example, if we have the expression $2^4$, we can take the logarithm of it to get $\log_2(2^4) = 4$.

Q: How do I use logarithms in real-world applications?

A: Logarithms are used in a variety of real-world applications, including:

• Finance: Logarithms are used to calculate interest rates and investment returns.
• Science: Logarithms are used to calculate the pH of a solution and the magnitude of an earthquake.
• Engineering: Logarithms are used to calculate the decibel level of a sound and the magnitude of a signal.
• Computer Science: Logarithms are used to calculate the time complexity of an algorithm and the space complexity of a data structure.

Q: What are some common logarithmic identities?

A: Some common logarithmic identities include:

• Product rule: $\log_a(b \cdot c) = \log_a(b) + \log_a(c)$
• Power rule: $\log_a(b^c) = c \cdot \log_a(b)$
• Quotient rule: $\log_a(\frac{b}{c}) = \log_a(b) - \log_a(c)$
• Change of base formula: $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$

Q: How do I use a calculator to evaluate a logarithmic expression?

A: To use a calculator to evaluate a logarithmic expression, follow these steps:

1. Enter the expression into the calculator.
2. Select the logarithmic function (e.g. $\log_2(x)$).
3. Enter the value of the argument (e.g. $16$).
4. Press the "Enter" or "Calculate" button.
5. The calculator will display the value of the expression.

Conclusion

In this article, we answered some frequently asked questions about simplifying logarithmic expressions. We covered topics such as the power rule and product rule of logarithms, evaluating logarithmic expressions, and using logarithms in real-world applications. We also provided some common logarithmic identities and explained how to use a calculator to evaluate a logarithmic expression.