For The Following Equation, Find $x$.$\log_b X = \frac{1}{2} \log_b 9 + \frac{2}{3} \log_b 125 - \log_b 15$

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be broken down into manageable steps. In this article, we will focus on solving the equation $\log_b x = \frac{1}{2} \log_b 9 + \frac{2}{3} \log_b 125 - \log_b 15$. We will use properties of logarithms to simplify the equation and isolate the variable $x$.

Understanding the Properties of Logarithms

Before we dive into solving the equation, it's essential to understand the properties of logarithms. The logarithm of a number $x$ with base $b$ is denoted as $\log_b x$. The logarithm of a product can be expressed as the sum of logarithms, and the logarithm of a quotient can be expressed as the difference of logarithms. These properties are essential in simplifying logarithmic equations.

Applying the Properties of Logarithms

To simplify the given equation, we can use the properties of logarithms to rewrite the equation in a more manageable form. We can start by using the property $\log_b (xy) = \log_b x + \log_b y$ to rewrite the equation as:

$\log_b x = \log_b (9^{\frac{1}{2}}) + \log_b (125^{\frac{2}{3}}) - \log_b (15)$

Simplifying the Equation

Now that we have rewritten the equation, we can simplify it further by evaluating the logarithms of the numbers inside the parentheses. We can use the property $\log_b x^n = n \log_b x$ to rewrite the equation as:

$\log_b x = \frac{1}{2} \log_b 9 + \frac{2}{3} \log_b 125 - \log_b 15$

Using the Change of Base Formula

To simplify the equation further, we can use the change of base formula, which states that $\log_b x = \frac{\log_a x}{\log_a b}$ for any positive real numbers $a$, $b$, and $x$. We can choose a convenient base, such as base 10, to simplify the equation.

Simplifying the Equation Using the Change of Base Formula

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

$\log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{1}{2} \frac{\log_{10} 9}{\log_{10} b} + \frac{2}{3} \frac{\log_{10} 125}{\log_{10} b} - \frac{\log_{10} 15}{\log_{10} b}$

Combining the Terms

Now that we have rewritten the equation using the change of base formula, we can combine the terms to simplify the equation further. We can use the property $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ to combine the terms.

Simplifying the Equation

Using the property $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$, we can rewrite the equation as:

$\log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{\frac{1}{2} \log_{10} 9 + \frac{2}{3} \log_{10} 125 - \log_{10} 15}{\log_{10} b}$

Evaluating the Logarithms

Now that we have simplified the equation, we can evaluate the logarithms to find the value of $x$. We can use a calculator to find the values of the logarithms.

Finding the Value of $x$

Using a calculator, we can find the values of the logarithms and simplify the equation further. We can use the property $\log_b x = \frac{\log_{10} x}{\log_{10} b}$ to find the value of $x$.

Conclusion

In this article, we have solved the logarithmic equation $\log_b x = \frac{1}{2} \log_b 9 + \frac{2}{3} \log_b 125 - \log_b 15$ using properties of logarithms and the change of base formula. We have simplified the equation step by step and evaluated the logarithms to find the value of $x$. The final answer is $\boxed{5}$.

Final Answer

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

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Rewrite the equation using the properties of logarithms: $\log_b x = \log_b (9^{\frac{1}{2}}) + \log_b (125^{\frac{2}{3}}) - \log_b (15)$
2. Simplify the equation: $\log_b x = \frac{1}{2} \log_b 9 + \frac{2}{3} \log_b 125 - \log_b 15$
3. Use the change of base formula: $\log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{1}{2} \frac{\log_{10} 9}{\log_{10} b} + \frac{2}{3} \frac{\log_{10} 125}{\log_{10} b} - \frac{\log_{10} 15}{\log_{10} b}$
4. Combine the terms: $\log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{\frac{1}{2} \log_{10} 9 + \frac{2}{3} \log_{10} 125 - \log_{10} 15}{\log_{10} b}$
5. Evaluate the logarithms: $\log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{\log_{10} 5}{\log_{10} b}$
6. Find the value of $x$: $x = b^{\log_{10} 5}$

Code Solution

Here is the code solution to the problem in Python:

import math

def solve_log_equation():
    # Define the variables
    b = 10  # base
    x = 5  # value of x

    # Calculate the logarithms
    log_x = math.log(x, b)
    log_9 = math.log(9, b)
    log_125 = math.log(125, b)
    log_15 = math.log(15, b)

    # Simplify the equation
    log_x = (1/2) * log_9 + (2/3) * log_125 - log_15

    # Find the value of x
    x = b ** log_x

    return x

# Execute the function
x = solve_log_equation()
print(x)

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

Introduction

In our previous article, we solved the logarithmic equation $\log_b x = \frac{1}{2} \log_b 9 + \frac{2}{3} \log_b 125 - \log_b 15$. In this article, we will answer some frequently asked questions related to logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves logarithms. It is a mathematical statement that contains a logarithmic expression, which is an expression that involves the logarithm of a number.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

• $\log_b (xy) = \log_b x + \log_b y$
• $\log_b (x/y) = \log_b x - \log_b y$
• $\log_b x^n = n \log_b x$

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you can use the properties of logarithms to rewrite the equation in a more manageable form. You can also use the change of base formula to simplify the equation.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows you to change the base of a logarithm. It is given by:

$\log_b x = \frac{\log_a x}{\log_a b}$

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use a calculator or a logarithmic table. You can also use the properties of logarithms to simplify the expression.

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

A: A logarithmic equation is an equation that involves logarithms, while an exponential equation is an equation that involves exponents. For example, the equation $\log_b x = 2$ is a logarithmic equation, while the equation $b^x = 100$ is an exponential equation.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithms to rewrite the equation in a more manageable form. You can also use the change of base formula to simplify the equation.

Q: What are some common logarithmic equations?

A: Some common logarithmic equations include:

• $\log_b x = 2$
• $\log_b x = \log_b 9 + \log_b 16$
• $\log_b x = \log_b 25 - \log_b 5$

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or a graphing software. You can also use the properties of logarithms to simplify the function.

Q: What are some real-world applications of logarithmic equations?

A: Logarithmic equations have many real-world applications, including:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design and optimize systems.

Conclusion

In this article, we have answered some frequently asked questions related to logarithmic equations. We have also provided some examples and real-world applications of logarithmic equations. We hope that this article has been helpful in understanding logarithmic equations.

Final Answer

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

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Rewrite the equation using the properties of logarithms: $\log_b x = \log_b (9^{\frac{1}{2}}) + \log_b (125^{\frac{2}{3}}) - \log_b (15)$
2. Simplify the equation: $\log_b x = \frac{1}{2} \log_b 9 + \frac{2}{3} \log_b 125 - \log_b 15$
3. Use the change of base formula: $\log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{1}{2} \frac{\log_{10} 9}{\log_{10} b} + \frac{2}{3} \frac{\log_{10} 125}{\log_{10} b} - \frac{\log_{10} 15}{\log_{10} b}$
4. Combine the terms: $\log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{\frac{1}{2} \log_{10} 9 + \frac{2}{3} \log_{10} 125 - \log_{10} 15}{\log_{10} b}$
5. Evaluate the logarithms: $\log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{\log_{10} 5}{\log_{10} b}$
6. Find the value of $x$: $x = b^{\log_{10} 5}$

Code Solution

Here is the code solution to the problem in Python:

import math

def solve_log_equation():
    # Define the variables
    b = 10  # base
    x = 5  # value of x

    # Calculate the logarithms
    log_x = math.log(x, b)
    log_9 = math.log(9, b)
    log_125 = math.log(125, b)
    log_15 = math.log(15, b)

    # Simplify the equation
    log_x = (1/2) * log_9 + (2/3) * log_125 - log_15

    # Find the value of x
    x = b ** log_x

    return x

# Execute the function
x = solve_log_equation()
print(x)

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