Simplify The Expression: $\[ 2 \log _3 5 - \log _3 10 + 3 \log _3 4 \\]

Introduction

In this article, we will simplify the given expression involving logarithms. The expression is ${ 2 \log _3 5 - \log _3 10 + 3 \log _3 4 }$ and we will use the properties of logarithms to simplify it.

Understanding Logarithms

Before we start simplifying the expression, let's understand what logarithms are. A logarithm is the inverse operation of exponentiation. In other words, if we have a number $x$ and we want to find the power to which a base number $b$ must be raised to get $x$, then the logarithm of $x$ with base $b$ is the exponent. For example, if we have $3^2 = 9$, then the logarithm of 9 with base 3 is 2.

Properties of Logarithms

There are several properties of logarithms that we will use to simplify the expression. These properties are:

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

Simplifying the Expression

Now that we have understood the properties of logarithms, let's simplify the expression. We will start by using the power rule to simplify the first term:

$${ 2 \log _3 5 = \log _3 5^2 = \log _3 25 }$$

Next, we will use the quotient rule to simplify the second term:

$${ \log _3 10 = \log _3 \frac{5 \cdot 2}{1} = \log _3 5 + \log _3 2 }$$

Now, we will use the product rule to simplify the third term:

$${ 3 \log _3 4 = \log _3 4^3 = \log _3 64 }$$

Combining the Terms

Now that we have simplified each term, let's combine them:

$${ \log _3 25 - \log _3 5 - \log _3 2 + \log _3 64 }$$

Using the Quotient Rule Again

We can use the quotient rule again to simplify the expression:

$${ \log _3 25 - \log _3 5 - \log _3 2 + \log _3 64 = \log _3 \frac{25}{5} - \log _3 2 + \log _3 64 }$$

$${ = \log _3 5 - \log _3 2 + \log _3 64 }$$

Using the Product Rule Again

We can use the product rule again to simplify the expression:

$${ \log _3 5 - \log _3 2 + \log _3 64 = \log _3 (5 \cdot 64) - \log _3 2 }$$

$${ = \log _3 320 - \log _3 2 }$$

Using the Quotient Rule Again

We can use the quotient rule again to simplify the expression:

$${ \log _3 320 - \log _3 2 = \log _3 \frac{320}{2} }$$

$${ = \log _3 160 }$$

Conclusion

In this article, we simplified the given expression involving logarithms. We used the properties of logarithms, such as the product rule, quotient rule, and power rule, to simplify the expression. The final simplified expression is $\log _3 160$.

Final Answer

The final answer is $\boxed{\log _3 160}$.

Frequently Asked Questions

• What is the product rule of logarithms?
  • The product rule of logarithms states that $\log_b (xy) = \log_b x + \log_b y$.
• What is the quotient rule of logarithms?
  • The quotient rule of logarithms states that $\log_b \frac{x}{y} = \log_b x - \log_b y$.
• What is the power rule of logarithms?
  • The power rule of logarithms states that $\log_b x^y = y \log_b x$.

References

• [1] "Logarithms." Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/x2f6b7f0f-logarithms.

Related Articles

• Simplifying Expressions Involving Logarithms
• Properties of Logarithms
  

Introduction

In our previous article, we simplified the expression ${ 2 \log _3 5 - \log _3 10 + 3 \log _3 4 }$ using the properties of logarithms. In this article, we will answer some frequently asked questions related to simplifying expressions involving logarithms.

Q&A

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log_b (xy) = \log_b x + \log_b y$. 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 $\log_b \frac{x}{y} = \log_b x - \log_b y$. 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 $\log_b x^y = y \log_b x$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I simplify an expression involving logarithms?

A: To simplify an expression involving logarithms, you can use the properties of logarithms, such as the product rule, quotient rule, and power rule. You can also use the fact that $\log_b 1 = 0$ and $\log_b b = 1$.

Q: What is the difference between a logarithm and an exponential function?

A: A logarithm is the inverse operation of an exponential function. In other words, if we have a number $x$ and we want to find the power to which a base number $b$ must be raised to get $x$, then the logarithm of $x$ with base $b$ is the exponent. For example, if we have $3^2 = 9$, then the logarithm of 9 with base 3 is 2.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use the fact that $\log_b x = y$ means that $b^y = x$. You can also use a calculator to evaluate a logarithmic expression.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all positive real numbers. In other words, the input of a logarithmic function must be a positive number.

Q: What is the range of a logarithmic function?

A: The range of a logarithmic function is all real numbers. In other words, the output of a logarithmic function can be any real number.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions involving logarithms. We hope that this article has been helpful in understanding the properties of logarithms and how to simplify expressions involving logarithms.

Final Answer

The final answer is $\boxed{\log _3 160}$.

Frequently Asked Questions

• What is the product rule of logarithms?
  • The product rule of logarithms states that $\log_b (xy) = \log_b x + \log_b y$.
• What is the quotient rule of logarithms?
  • The quotient rule of logarithms states that $\log_b \frac{x}{y} = \log_b x - \log_b y$.
• What is the power rule of logarithms?
  • The power rule of logarithms states that $\log_b x^y = y \log_b x$.

References

• [1] "Logarithms." Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/x2f6b7f0f-logarithms.

Related Articles

• Simplifying Expressions Involving Logarithms
• Properties of Logarithms