Simplify The Expression: $\ln 2 + \ln 8 - \ln 4$Options:A. $\ln 4$B. $\ln 6$C. $\ln 64$

Feb 25, 2025 by ADMIN 88 views

Simplify the Expression: $\ln 2 + \ln 8 - \ln 4$

In this article, we will simplify the given expression $\ln 2 + \ln 8 - \ln 4$ . This expression involves logarithmic functions, and we will use the properties of logarithms to simplify it. We will also explore the concept of logarithms and their properties in detail.

A logarithm is the inverse operation of exponentiation. It is a mathematical function that takes a number as input and returns a value that represents the power to which a base number must be raised to produce the input number. In other words, if $x = \log_b a$ , then $b^x = a$ . The logarithm of a number is a measure of its magnitude or size.

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

Product Property: $\log_b (xy) = \log_b x + \log_b y$
Quotient Property: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
Power Property: $\log_b x^y = y \log_b x$

Now, let's simplify the given expression $\ln 2 + \ln 8 - \ln 4$ using the properties of logarithms.

$\ln 2 + \ln 8 - \ln 4$

Using the product property, we can rewrite $\ln 8$ as $\ln (2^3)$ .

$\ln 2 + \ln (2^3) - \ln 4$

Using the power property, we can rewrite $\ln (2^3)$ as $3 \ln 2$ .

$\ln 2 + 3 \ln 2 - \ln 4$

Using the quotient property, we can rewrite $\ln 4$ as $\ln (2^2)$ .

$\ln 2 + 3 \ln 2 - \ln (2^2)$

Using the power property, we can rewrite $\ln (2^2)$ as $2 \ln 2$ .

$\ln 2 + 3 \ln 2 - 2 \ln 2$

Now, we can combine the terms.

$\ln 2 + \ln 2 + \ln 2 - 2 \ln 2$

$\ln 2$

Therefore, the simplified expression is $\ln 2$ .

In this article, we simplified the expression $\ln 2 + \ln 8 - \ln 4$ using the properties of logarithms. We used the product property, quotient property, and power property to rewrite the expression and simplify it. The final simplified expression is $\ln 2$ . We hope this article has provided a clear understanding of how to simplify logarithmic expressions using the properties of logarithms.

The final answer is $\boxed{\ln 2}$ .

A. $\ln 4$ B. $\ln 6$ C. $\ln 64$

The correct option is A. $\ln 4$ is incorrect, B. $\ln 6$ is incorrect, and C. $\ln 64$ is incorrect. The correct answer is $\boxed{\ln 2}$ .
Simplify the Expression: $\ln 2 + \ln 8 - \ln 4$ - Q&A

In our previous article, we simplified the expression $\ln 2 + \ln 8 - \ln 4$ using the properties of logarithms. We used the product property, quotient property, and power property to rewrite the expression and simplify it. The final simplified expression is $\ln 2$ . In this article, we will answer some frequently asked questions related to the simplification of logarithmic expressions.

Q: What is the product property of logarithms?

A: The product property 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 property of logarithms?

A: The quotient property of logarithms states that $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$ . 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 property of logarithms?

A: The power property 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 a logarithmic expression using the properties of logarithms?

A: To simplify a logarithmic expression using the properties of logarithms, you need to identify the properties that can be applied to the expression. You can use the product property, quotient property, and power property to rewrite the expression and simplify it.

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

A: A logarithmic expression is an expression that involves a logarithm, which is the inverse operation of exponentiation. An exponential expression is an expression that involves an exponent, which is the power to which a base number is raised.

Q: Can I simplify a logarithmic expression using algebraic manipulations?

A: Yes, you can simplify a logarithmic expression using algebraic manipulations. However, you need to be careful when using algebraic manipulations, as they may not always preserve the properties of logarithms.

Q: What is the final answer to the expression $\ln 2 + \ln 8 - \ln 4$ ?

A: The final answer to the expression $\ln 2 + \ln 8 - \ln 4$ is $\ln 2$ .

In this article, we answered some frequently asked questions related to the simplification of logarithmic expressions. We discussed the product property, quotient property, and power property of logarithms, and provided examples of how to simplify logarithmic expressions using these properties. We also discussed the difference between logarithmic expressions and exponential expressions, and provided tips on how to simplify logarithmic expressions using algebraic manipulations.