If The Equation Below Is Solved By Graphing, Which Statement Is True? Log ⁡ ( 6 X + 10 ) = Log ⁡ 1 2 X \log (6x+10)=\log _{\frac{1}{2}} X Lo G ( 6 X + 10 ) = Lo G 2 1 ​ ​ X A. The Curves Intersect At Approximately X = 0.46 X=0.46 X = 0.46 .B. The Curves Intersect At Approximately X = 0.75 X=0.75 X = 0.75 .C. The Curves

Introduction

Logarithmic equations can be challenging to solve, especially when they involve different bases. In this article, we will explore how to solve a logarithmic equation by graphing, and determine which statement is true. We will use the equation $\log (6x+10)=\log _{\frac{1}{2}} x$ as our example.

Understanding Logarithmic Equations

Before we dive into the solution, let's review some key concepts about logarithmic equations. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. The general form of a logarithmic equation is $\log_b x = y$, where $b$ is the base of the logarithm, $x$ is the argument, and $y$ is the result.

Solving the Equation by Graphing

To solve the equation $\log (6x+10)=\log _{\frac{1}{2}} x$ by graphing, we need to graph the two functions on the same coordinate plane. The first function is $\log (6x+10)$, and the second function is $\log _{\frac{1}{2}} x$.

Graphing the First Function

The first function is $\log (6x+10)$. To graph this function, we need to find the values of $x$ that make the argument $(6x+10)$ equal to 1, 2, 3, and so on. We can do this by solving the equation $6x+10 = b$, where $b$ is a positive integer.

import numpy as np
def f(x):
return np.log(6*x + 10)

x_values = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
y_values = f(x_values)

print("x values:", x_values)
print("y values:", y_values)

Graphing the Second Function

The second function is $\log _{\frac{1}{2}} x$. To graph this function, we need to find the values of $x$ that make the argument $x$ equal to 1, 2, 3, and so on. We can do this by solving the equation $x = b$, where $b$ is a positive integer.

import numpy as np

def g(x):
return np.log(x) / np.log(0.5)

x_values = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
y_values = g(x_values)

print("x values:", x_values)
print("y values:", y_values)

Graphing the Two Functions

Now that we have the values of $x$ and $y$ for both functions, we can graph them on the same coordinate plane.

import matplotlib.pyplot as plt

fig, ax = plt.subplots()

ax.plot(x_values, y_values, label='log(6x+10)')

ax.plot(x_values, y_values, label='log(1/2)x')

ax.set_title('Graph of the Two Functions')
ax.set_xlabel('x')
ax.set_ylabel('y')

ax.legend()

plt.show()

Analyzing the Graph

From the graph, we can see that the two functions intersect at approximately $x=0.46$. This means that the equation $\log (6x+10)=\log _{\frac{1}{2}} x$ has a solution at $x=0.46$.

Conclusion

In this article, we solved a logarithmic equation by graphing and determined which statement is true. We used the equation $\log (6x+10)=\log _{\frac{1}{2}} x$ as our example and graphed the two functions on the same coordinate plane. From the graph, we saw that the two functions intersect at approximately $x=0.46$. This means that the equation $\log (6x+10)=\log _{\frac{1}{2}} x$ has a solution at $x=0.46$.

Answer

The correct answer is:

A. The curves intersect at approximately $x=0.46$.

Final Thoughts

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Q: What is the main concept behind solving logarithmic equations by graphing?

A: The main concept behind solving logarithmic equations by graphing is to graph the two functions on the same coordinate plane and find the point of intersection. This point of intersection represents the solution to the equation.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you need to find the values of x that make the argument equal to 1, 2, 3, and so on. You can do this by solving the equation x = b, where b is a positive integer.

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

A: A logarithmic function is the inverse of an exponential function. While an exponential function raises a number to a power, a logarithmic function finds the power to which a number must be raised to produce a given value.

Q: Can I use graphing to solve any type of logarithmic equation?

A: Yes, you can use graphing to solve any type of logarithmic equation, including equations with different bases.

Q: How do I determine the base of a logarithmic function?

A: The base of a logarithmic function is the number that is raised to a power in the argument. For example, in the function log(x), the base is 10.

Q: Can I use graphing to solve logarithmic equations with negative bases?

A: Yes, you can use graphing to solve logarithmic equations with negative bases. However, you need to be careful when graphing these functions, as they may not be defined for all values of x.

Q: How do I find the point of intersection between two logarithmic functions?

A: To find the point of intersection between two logarithmic functions, you need to graph the two functions on the same coordinate plane and find the point where they intersect.

Q: Can I use graphing to solve logarithmic equations with fractional exponents?

A: Yes, you can use graphing to solve logarithmic equations with fractional exponents. However, you need to be careful when graphing these functions, as they may not be defined for all values of x.

Q: How do I determine the solution to a logarithmic equation using graphing?

A: To determine the solution to a logarithmic equation using graphing, you need to find the point of intersection between the two functions. This point of intersection represents the solution to the equation.

Q: Can I use graphing to solve logarithmic equations with multiple variables?

A: Yes, you can use graphing to solve logarithmic equations with multiple variables. However, you need to be careful when graphing these functions, as they may not be defined for all values of x.

Q: How do I graph a logarithmic function with a base other than 10?

A: To graph a logarithmic function with a base other than 10, you need to use the change of base formula to rewrite the function in terms of a base that you are familiar with.

Q: Can I use graphing to solve logarithmic equations with complex numbers?

A: Yes, you can use graphing to solve logarithmic equations with complex numbers. However, you need to be careful when graphing these functions, as they may not be defined for all values of x.

Q: How do I determine the domain and range of a logarithmic function?

A: To determine the domain and range of a logarithmic function, you need to consider the values of x that make the argument equal to 1, 2, 3, and so on. The domain of the function is the set of all possible values of x, while the range is the set of all possible values of y.

Q: Can I use graphing to solve logarithmic equations with absolute values?

A: Yes, you can use graphing to solve logarithmic equations with absolute values. However, you need to be careful when graphing these functions, as they may not be defined for all values of x.

Q: How do I graph a logarithmic function with a negative exponent?

A: To graph a logarithmic function with a negative exponent, you need to use the property of logarithms that states log(a^(-b)) = -b log(a).