Solve The Equation: Log ⁡ M 9 + Log ⁡ ( M 2 ) 3 = 5 2 \log _m 9 + \log _{\left(m^2\right)} 3 = \frac{5}{2} Lo G M ​ 9 + Lo G ( M 2 ) ​ 3 = 2 5 ​

$$

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will focus on solving the equation $\log _m 9 + \log _{\left(m^2\right)} 3 = \frac{5}{2}$. This equation involves logarithms with different bases, and we will use various properties of logarithms to simplify and solve it.

Understanding the Equation

The given equation is $\log _m 9 + \log _{\left(m^2\right)} 3 = \frac{5}{2}$. To solve this equation, we need to understand the properties of logarithms. The logarithm of a number $x$ with base $b$ is denoted by $\log _b x$. It is the exponent to which the base $b$ must be raised to produce the number $x$. For example, $\log _2 8 = 3$ because $2^3 = 8$.

Using the Change of Base Formula

One of the key properties of logarithms is the change of base formula, which states that $\log _b x = \frac{\log _c x}{\log _c b}$ for any positive real numbers $b$, $c$, and $x$. We can use this formula to rewrite the given equation in a more manageable form.

Applying the Change of Base Formula

Using the change of base formula, we can rewrite the equation as follows:

$$\frac{\log 9}{\log m} + \frac{\log 3}{\log m^2} = \frac{5}{2}$$

Simplifying the Equation

We can simplify the equation by combining the fractions:

$$\frac{\log 9 + \log 3}{\log m} = \frac{5}{2}$$

Using the Product Rule for Logarithms

The product rule for logarithms states that $\log (xy) = \log x + \log y$. We can use this rule to simplify the numerator of the fraction:

$$\frac{\log (9 \cdot 3)}{\log m} = \frac{5}{2}$$

Simplifying the Numerator

Using the product rule for logarithms, we can simplify the numerator:

$$\frac{\log 27}{\log m} = \frac{5}{2}$$

Using the Power Rule for Logarithms

The power rule for logarithms states that $\log (x^y) = y \log x$. We can use this rule to simplify the numerator:

$$\frac{\log m^3}{\log m} = \frac{5}{2}$$

Simplifying the Fraction

Using the power rule for logarithms, we can simplify the fraction:

$$3 = \frac{5}{2}$$

Solving for $m$

We can solve for $m$ by multiplying both sides of the equation by $\frac{2}{5}$:

$$\frac{6}{5}m = 1$$

Solving for $m$

We can solve for $m$ by multiplying both sides of the equation by $\frac{5}{6}$:

$$m = \frac{5}{6}$$

Conclusion

In this article, we solved the logarithmic equation $\log _m 9 + \log _{\left(m^2\right)} 3 = \frac{5}{2}$. We used various properties of logarithms, including the change of base formula, the product rule, and the power rule, to simplify and solve the equation. The solution to the equation is $m = \frac{5}{6}$.

Final Answer

The final answer is $\boxed{\frac{5}{6}}$.

$$

Introduction

In our previous article, we solved the logarithmic equation $\log _m 9 + \log _{\left(m^2\right)} 3 = \frac{5}{2}$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on solving logarithmic equations.

Q: What is the change of base formula?

A: The change of base formula is a property of logarithms that allows us to rewrite a logarithm with a different base. It states that $\log _b x = \frac{\log _c x}{\log _c b}$ for any positive real numbers $b$, $c$, and $x$.

Q: How do I apply the change of base formula?

A: To apply the change of base formula, you need to identify the base of the logarithm and the new base you want to use. Then, you can rewrite the logarithm using the formula: $\frac{\log x}{\log b}$.

Q: What is the product rule for logarithms?

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

Q: How do I use the product rule for logarithms?

A: To use the product rule for logarithms, you need to identify the product inside the logarithm and rewrite it as the sum of the logarithms of the individual numbers.

Q: What is the power rule for logarithms?

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

Q: How do I use the power rule for logarithms?

A: To use the power rule for logarithms, you need to identify the power inside the logarithm and rewrite it as the exponent multiplied by the logarithm of the base.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to use the properties of logarithms, such as the change of base formula, the product rule, and the power rule, to simplify the equation and isolate the variable.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include:

• Not using the correct properties of logarithms
• Not simplifying the equation enough
• Not isolating the variable correctly
• Not checking the solution for validity

Q: How do I check the solution for validity?

A: To check the solution for validity, you need to plug the solution back into the original equation and verify that it is true.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide additional information on solving logarithmic equations. We hope that this article has been helpful in understanding the properties of logarithms and how to solve logarithmic equations.

Final Answer

The final answer is $\boxed{\frac{5}{6}}$.