Rewrite The Logarithmic Expression As A Single Logarithm With The Same Base.Simplify Any Fractions.${ \log 6 + 4 \log 2 + \log 3 }${ \square\$}

Introduction

In this article, we will explore the concept of rewriting logarithmic expressions as a single logarithm with the same base. This is an essential skill in mathematics, particularly in algebra and calculus. We will use the properties of logarithms to simplify the given expression and provide a clear understanding of the process.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. It is a mathematical function that takes a number as input and returns the power to which the base must be raised to produce that number. For example, if we have the equation $2^3 = 8$, then the logarithm of 8 with base 2 is 3, denoted as $\log_2 8 = 3$.

Properties of Logarithms

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

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

Rewriting the Logarithmic Expression

The given expression is $\log 6 + 4 \log 2 + \log 3$. We can rewrite this expression using the properties of logarithms.

Step 1: Apply the Power Property

We can rewrite the expression $4 \log 2$ using the power property. Since $4 = 2^2$, we can rewrite the expression as $2 \log 2^2$. Using the power property, we can rewrite this as $2 \cdot 2 \log 2 = 4 \log 2$.

\log 6 + 4 \log 2 + \log 3
= \log 6 + 2 \log 2^2 + \log 3
= \log 6 + 2 \cdot 2 \log 2 + \log 3
= \log 6 + 4 \log 2 + \log 3

Step 2: Apply the Product Property

We can rewrite the expression $\log 6 + \log 3$ using the product property. Since $\log 6 = \log (2 \cdot 3)$, we can rewrite the expression as $\log (2 \cdot 3) + \log 3$. Using the product property, we can rewrite this as $\log (2 \cdot 3 \cdot 3) = \log (2 \cdot 3^2)$.

\log 6 + \log 3
= \log (2 \cdot 3) + \log 3
= \log (2 \cdot 3 \cdot 3)
= \log (2 \cdot 3^2)

Step 3: Simplify the Expression

We can simplify the expression $\log 6 + 4 \log 2 + \log 3$ by combining the terms. Using the product property, we can rewrite the expression as $\log (2 \cdot 3^2) + 4 \log 2$. Using the product property again, we can rewrite this as $\log (2 \cdot 3^2 \cdot 2^4) = \log (2^5 \cdot 3^2)$.

\log 6 + 4 \log 2 + \log 3
= \log (2 \cdot 3^2) + 4 \log 2
= \log (2 \cdot 3^2 \cdot 2^4)
= \log (2^5 \cdot 3^2)

Step 4: Simplify the Fraction

We can simplify the fraction $\log (2^5 \cdot 3^2)$ by using the product property. Since $\log (2^5 \cdot 3^2) = \log 2^5 + \log 3^2$, we can rewrite the expression as $5 \log 2 + 2 \log 3$.

\log (2^5 \cdot 3^2)
= \log 2^5 + \log 3^2
= 5 \log 2 + 2 \log 3

Conclusion

In this article, we have rewritten the logarithmic expression $\log 6 + 4 \log 2 + \log 3$ as a single logarithm with the same base. We have used the properties of logarithms to simplify the expression and provide a clear understanding of the process. The final expression is $5 \log 2 + 2 \log 3$, which is a single logarithm with the same base.

Final Answer

Introduction

In our previous article, we explored the concept of rewriting logarithmic expressions as a single logarithm with the same base. We used the properties of logarithms to simplify the given expression and provide a clear understanding of the process. In this article, we will answer some frequently asked questions related to rewriting logarithmic expressions as a single logarithm with the same base.

Q: What are the properties of logarithms that we can use to rewrite logarithmic expressions?

A: There are several properties of logarithms that we can use to rewrite logarithmic expressions. These properties are:

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

Q: How do we apply the product property to rewrite logarithmic expressions?

A: To apply the product property, we can rewrite the expression as the sum of two logarithms. For example, if we have the expression $\log (2 \cdot 3)$, we can rewrite it as $\log 2 + \log 3$.

\log (2 \cdot 3)
= \log 2 + \log 3

Q: How do we apply the quotient property to rewrite logarithmic expressions?

A: To apply the quotient property, we can rewrite the expression as the difference of two logarithms. For example, if we have the expression $\log \frac{2}{3}$, we can rewrite it as $\log 2 - \log 3$.

\log \frac{2}{3}
= \log 2 - \log 3

Q: How do we apply the power property to rewrite logarithmic expressions?

A: To apply the power property, we can rewrite the expression as the product of the exponent and the logarithm. For example, if we have the expression $\log 2^3$, we can rewrite it as $3 \log 2$.

\log 2^3
= 3 \log 2

Q: Can we rewrite logarithmic expressions with different bases as a single logarithm with the same base?

A: Yes, we can rewrite logarithmic expressions with different bases as a single logarithm with the same base. We can use the change of base formula to rewrite the expression. The change of base formula is $\log_a x = \frac{\log_b x}{\log_b a}$.

\log_a x
= \frac{\log_b x}{\log_b a}

Q: How do we simplify fractions in logarithmic expressions?

A: To simplify fractions in logarithmic expressions, we can use the properties of logarithms. We can rewrite the fraction as the difference of two logarithms and then simplify the expression.

\log \frac{x}{y}
= \log x - \log y

Conclusion

In this article, we have answered some frequently asked questions related to rewriting logarithmic expressions as a single logarithm with the same base. We have used the properties of logarithms to simplify the expressions and provide a clear understanding of the process. We hope that this article has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer is $\boxed{5 \log 2 + 2 \log 3}$.