Given That:$\[ \begin{align*} \log_{10} 2 &= 0.3010 \\ \log_{10} 3 &= 0.4771 \\ \log_{10} 5 &= 0.6990 \end{align*} \\]Find \[$\log_{10} 1.5 + \log_{10} 45\$\].

Introduction

Logarithms are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and computer science. In this article, we will explore the properties of logarithms and how they can be used to simplify complex expressions. We will also apply these properties to solve a specific problem involving logarithmic expressions.

Logarithmic Properties

Before we dive into the problem, let's review some basic logarithmic properties:

• Product Rule: $\log_{a} (xy) = \log_{a} x + \log_{a} y$
• Quotient Rule: $\log_{a} \left(\frac{x}{y}\right) = \log_{a} x - \log_{a} y$
• Power Rule: $\log_{a} x^{y} = y \log_{a} x$

These properties will be essential in solving the problem at hand.

The Problem

Given the logarithmic values:

{ \begin{align*} \log_{10} 2 &= 0.3010 \\ \log_{10} 3 &= 0.4771 \\ \log_{10} 5 &= 0.6990 \end{align*} \}

We need to find the value of $\log_{10} 1.5 + \log_{10} 45$.

Step 1: Simplify the Expression

To simplify the expression, we can use the product rule, which states that $\log_{a} (xy) = \log_{a} x + \log_{a} y$. In this case, we can rewrite the expression as:

$\log_{10} (1.5 \times 45)$

Now, we can use the product rule to simplify the expression:

$\log_{10} 1.5 + \log_{10} 45 = \log_{10} (1.5 \times 45)$

Step 2: Evaluate the Expression

To evaluate the expression, we need to find the value of $\log_{10} (1.5 \times 45)$. We can do this by multiplying the values of $\log_{10} 1.5$ and $\log_{10} 45$.

However, we don't have the value of $\log_{10} 1.5$ directly. We can find it by using the fact that $\log_{10} 1.5 = \log_{10} \left(\frac{3}{2}\right)$. We can rewrite this as:

$\log_{10} \left(\frac{3}{2}\right) = \log_{10} 3 - \log_{10} 2$

Now, we can substitute the values of $\log_{10} 3$ and $\log_{10} 2$:

$\log_{10} \left(\frac{3}{2}\right) = 0.4771 - 0.3010 = 0.1761$

Step 3: Find the Value of $\log_{10} 45$

To find the value of $\log_{10} 45$, we can use the fact that $\log_{10} 45 = \log_{10} (9 \times 5)$. We can rewrite this as:

$\log_{10} (9 \times 5) = \log_{10} 9 + \log_{10} 5$

Now, we can substitute the values of $\log_{10} 9$ and $\log_{10} 5$:

$\log_{10} (9 \times 5) = \log_{10} 9 + \log_{10} 5 = 0.9542 + 0.6990 = 1.6532$

Step 4: Find the Value of $\log_{10} 1.5 + \log_{10} 45$

Now that we have the values of $\log_{10} 1.5$ and $\log_{10} 45$, we can find the value of $\log_{10} 1.5 + \log_{10} 45$:

$\log_{10} 1.5 + \log_{10} 45 = 0.1761 + 1.6532 = 1.8293$

Conclusion

In this article, we used logarithmic properties to simplify a complex expression involving logarithmic values. We applied the product rule, quotient rule, and power rule to find the value of $\log_{10} 1.5 + \log_{10} 45$. The final answer is $\boxed{1.8293}$.

Logarithmic Properties

• Product Rule: $\log_{a} (xy) = \log_{a} x + \log_{a} y$
• Quotient Rule: $\log_{a} \left(\frac{x}{y}\right) = \log_{a} x - \log_{a} y$
• Power Rule: $\log_{a} x^{y} = y \log_{a} x$

Logarithmic Values

{ \begin{align*} \log_{10} 2 &= 0.3010 \\ \log_{10} 3 &= 0.4771 \\ \log_{10} 5 &= 0.6990 \end{align*} \}

The Problem

Find the value of $\log_{10} 1.5 + \log_{10} 45$.

Step 1: Simplify the Expression

$\log_{10} 1.5 + \log_{10} 45 = \log_{10} (1.5 \times 45)$

Step 2: Evaluate the Expression

$\log_{10} 1.5 + \log_{10} 45 = \log_{10} (1.5 \times 45) = 0.1761 + 1.6532 = 1.8293$

Conclusion

$$

Introduction

In our previous article, we explored the properties of logarithms and how they can be used to simplify complex expressions. We also applied these properties to solve a specific problem involving logarithmic expressions. In this article, we will answer some frequently asked questions about logarithmic properties and their applications.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log_{a} (xy) = \log_{a} x + \log_{a} y$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that $\log_{a} \left(\frac{x}{y}\right) = \log_{a} x - \log_{a} y$. This means that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log_{a} x^{y} = y \log_{a} x$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I apply the product rule of logarithms?

A: To apply the product rule of logarithms, you need to multiply the individual factors together and then take the logarithm of the result. For example, if you want to find the logarithm of $2 \times 3$, you would first multiply $2$ and $3$ together to get $6$, and then take the logarithm of $6$.

Q: How do I apply the quotient rule of logarithms?

A: To apply the quotient rule of logarithms, you need to divide the individual factors and then take the logarithm of the result. For example, if you want to find the logarithm of $\frac{2}{3}$, you would first divide $2$ by $3$ to get $\frac{2}{3}$, and then take the logarithm of $\frac{2}{3}$.

Q: How do I apply the power rule of logarithms?

A: To apply the power rule of logarithms, you need to multiply the exponent by the logarithm of the base. For example, if you want to find the logarithm of $2^{3}$, you would first multiply the exponent $3$ by the logarithm of $2$ to get $3 \log_{2} 2$, and then simplify the expression.

Q: What are some common logarithmic values?

A: Some common logarithmic values include:

• $\log_{10} 2 = 0.3010$
• $\log_{10} 3 = 0.4771$
• $\log_{10} 5 = 0.6990$
• $\log_{10} 10 = 1$

Q: How do I use logarithmic properties to simplify complex expressions?

A: To use logarithmic properties to simplify complex expressions, you need to identify the properties that can be applied to the expression. For example, if you have an expression like $\log_{10} (2 \times 3)$, you can use the product rule of logarithms to simplify the expression to $\log_{10} 2 + \log_{10} 3$.

Conclusion

In this article, we answered some frequently asked questions about logarithmic properties and their applications. We also provided examples of how to apply the product rule, quotient rule, and power rule of logarithms. By understanding these properties and how to apply them, you can simplify complex expressions and solve problems involving logarithmic values.

Logarithmic Properties

• Product Rule: $\log_{a} (xy) = \log_{a} x + \log_{a} y$
• Quotient Rule: $\log_{a} \left(\frac{x}{y}\right) = \log_{a} x - \log_{a} y$
• Power Rule: $\log_{a} x^{y} = y \log_{a} x$

Common Logarithmic Values

• $\log_{10} 2 = 0.3010$
• $\log_{10} 3 = 0.4771$
• $\log_{10} 5 = 0.6990$
• $\log_{10} 10 = 1$

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log_{a} (xy) = \log_{a} x + \log_{a} y$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that $\log_{a} \left(\frac{x}{y}\right) = \log_{a} x - \log_{a} y$. This means that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log_{a} x^{y} = y \log_{a} x$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I apply the product rule of logarithms?

A: To apply the product rule of logarithms, you need to multiply the individual factors together and then take the logarithm of the result. For example, if you want to find the logarithm of $2 \times 3$, you would first multiply $2$ and $3$ together to get $6$, and then take the logarithm of $6$.

Q: How do I apply the quotient rule of logarithms?

A: To apply the quotient rule of logarithms, you need to divide the individual factors and then take the logarithm of the result. For example, if you want to find the logarithm of $\frac{2}{3}$, you would first divide $2$ by $3$ to get $\frac{2}{3}$, and then take the logarithm of $\frac{2}{3}$.

Q: How do I apply the power rule of logarithms?

A: To apply the power rule of logarithms, you need to multiply the exponent by the logarithm of the base. For example, if you want to find the logarithm of $2^{3}$, you would first multiply the exponent $3$ by the logarithm of $2$ to get $3 \log_{2} 2$, and then simplify the expression.

Q: What are some common logarithmic values?

A: Some common logarithmic values include:

• $\log_{10} 2 = 0.3010$
• $\log_{10} 3 = 0.4771$
• $\log_{10} 5 = 0.6990$
• $\log_{10} 10 = 1$

Q: How do I use logarithmic properties to simplify complex expressions?

A: To use logarithmic properties to simplify complex expressions, you need to identify the properties that can be applied to the expression. For example, if you have an expression like $\log_{10} (2 \times 3)$, you can use the product rule of logarithms to simplify the expression to $\log_{10} 2 + \log_{10} 3$.

Conclusion

In this article, we answered some frequently asked questions about logarithmic properties and their applications. We also provided examples of how to apply the product rule, quotient rule, and power rule of logarithms. By understanding these properties and how to apply them, you can simplify complex expressions and solve problems involving logarithmic values.