What Is The Approximate Value Of $x$ In The Equation Below? Log ⁡ 3 4 25 = 3 X − 1 \log _{\frac{3}{4}} 25=3 X-1 Lo G 4 3 ​ ​ 25 = 3 X − 1 A. 0.304 B. 0.955 C. -3.396 D. -0.708

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Understanding the Problem

The given equation is $\log _{\frac{3}{4}} 25=3 x-1$. To find the approximate value of $x$, we need to solve for $x$ using logarithmic properties and algebraic manipulation.

Using Logarithmic Properties

The equation involves a logarithm with a base of $\frac{3}{4}$. To simplify the equation, we can use the property of logarithms that states $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive real number.

Converting to Common Logarithm

We can convert the given logarithm to a common logarithm by using the property mentioned above. Let's choose base 10 as our common logarithm.

$\log _{\frac{3}{4}} 25 = \frac{\log_{10} 25}{\log_{10} \frac{3}{4}}$

Evaluating the Logarithms

Now, we can evaluate the logarithms using a calculator or a logarithm table.

$\log_{10} 25 \approx 1.39794$

$\log_{10} \frac{3}{4} \approx -0.12494$

Substituting the Values

Substituting the values of the logarithms back into the equation, we get:

$\frac{1.39794}{-0.12494} = 3x - 1$

Simplifying the Equation

Simplifying the equation, we get:

$-11.21 = 3x - 1$

Solving for $x$

Adding 1 to both sides of the equation, we get:

$-10.21 = 3x$

Dividing both sides of the equation by 3, we get:

$x \approx -3.407$

Rounding the Value

Rounding the value of $x$ to three decimal places, we get:

$x \approx -3.407$

Comparing with the Options

Comparing the calculated value of $x$ with the given options, we can see that the closest value is:

C. -3.396

Conclusion

The approximate value of $x$ in the equation $\log _{\frac{3}{4}} 25=3 x-1$ is $-3.396$.

Final Answer

The final answer is $\boxed{-3.396}$.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. It is a mathematical expression that represents the power to which a base number must be raised to obtain a given value.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithmic term and then use the properties of logarithms to simplify the equation. You can use the property of logarithms that states $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive real number.

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

A: A logarithmic equation and an exponential equation are related but distinct concepts. An exponential equation is an equation that involves an exponential expression, such as $2^x = 8$. A logarithmic equation is an equation that involves a logarithmic expression, such as $\log_2 8 = x$.

Q: How do I convert a logarithmic equation to an exponential equation?

A: To convert a logarithmic equation to an exponential equation, you can use the property of logarithms that states $\log_b a = c$ is equivalent to $b^c = a$. For example, the logarithmic equation $\log_2 8 = x$ is equivalent to the exponential equation $2^x = 8$.

Q: What is the base of a logarithm?

A: The base of a logarithm is the number that is raised to a power to obtain a given value. For example, in the logarithmic equation $\log_2 8 = x$, the base is 2.

Q: How do I evaluate a logarithm?

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

Q: What is the difference between a common logarithm and a natural logarithm?

A: A common logarithm is a logarithm with a base of 10, while a natural logarithm is a logarithm with a base of $e$, where $e$ is a mathematical constant approximately equal to 2.71828.

Q: How do I use a calculator to evaluate a logarithm?

A: To use a calculator to evaluate a logarithm, you need to enter the base and the argument of the logarithm, and then press the logarithm button. For example, to evaluate the logarithm $\log_{10} 25$, you would enter 10 as the base and 25 as the argument, and then press the logarithm button.

Q: What is the significance of logarithmic equations in real-world applications?

A: Logarithmic equations have many real-world applications, including finance, science, and engineering. They are used to model population growth, chemical reactions, and electrical circuits, among other things.

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 and then graph it.

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

A: A logarithmic function and an exponential function are related but distinct concepts. A logarithmic function is a function that involves a logarithmic expression, such as $\log_2 x$. An exponential function is a function that involves an exponential expression, such as $2^x$.

Q: How do I use logarithmic equations to solve problems in finance?

A: Logarithmic equations can be used to solve problems in finance, such as calculating interest rates and returns on investment. They can also be used to model stock prices and other financial instruments.

Q: What is the significance of logarithmic equations in science and engineering?

A: Logarithmic equations have many applications in science and engineering, including modeling population growth, chemical reactions, and electrical circuits. They are also used to analyze data and make predictions.

Q: How do I use logarithmic equations to solve problems in computer science?

A: Logarithmic equations can be used to solve problems in computer science, such as analyzing algorithms and data structures. They can also be used to model network traffic and other computer systems.

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

A: A logarithmic equation and a polynomial equation are distinct concepts. A logarithmic equation is an equation that involves a logarithmic expression, such as $\log_2 8 = x$. A polynomial equation is an equation that involves a polynomial expression, such as $x^2 + 3x - 4 = 0$.

Q: How do I use logarithmic equations to solve problems in mathematics?

A: Logarithmic equations can be used to solve problems in mathematics, such as analyzing functions and making predictions. They can also be used to model mathematical concepts, such as fractals and chaos theory.

Q: What is the significance of logarithmic equations in mathematics education?

A: Logarithmic equations have many applications in mathematics education, including teaching algebra and calculus. They can also be used to model mathematical concepts and make predictions.

Q: How do I use logarithmic equations to solve problems in statistics?

A: Logarithmic equations can be used to solve problems in statistics, such as analyzing data and making predictions. They can also be used to model statistical concepts, such as probability and hypothesis testing.

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

A: A logarithmic equation and a trigonometric equation are distinct concepts. A logarithmic equation is an equation that involves a logarithmic expression, such as $\log_2 8 = x$. A trigonometric equation is an equation that involves a trigonometric expression, such as $\sin x = \frac{1}{2}$.

Q: How do I use logarithmic equations to solve problems in physics?

A: Logarithmic equations can be used to solve problems in physics, such as analyzing motion and making predictions. They can also be used to model physical concepts, such as energy and momentum.

Q: What is the significance of logarithmic equations in physics education?

A: Logarithmic equations have many applications in physics education, including teaching mechanics and electromagnetism. They can also be used to model physical concepts and make predictions.

Q: How do I use logarithmic equations to solve problems in engineering?

A: Logarithmic equations can be used to solve problems in engineering, such as analyzing systems and making predictions. They can also be used to model engineering concepts, such as control systems and signal processing.

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

A: A logarithmic equation and a differential equation are distinct concepts. A logarithmic equation is an equation that involves a logarithmic expression, such as $\log_2 8 = x$. A differential equation is an equation that involves a derivative, such as $\frac{dy}{dx} = 2x$.

Q: How do I use logarithmic equations to solve problems in economics?

A: Logarithmic equations can be used to solve problems in economics, such as analyzing data and making predictions. They can also be used to model economic concepts, such as supply and demand.

Q: What is the significance of logarithmic equations in economics education?

A: Logarithmic equations have many applications in economics education, including teaching microeconomics and macroeconomics. They can also be used to model economic concepts and make predictions.

Q: How do I use logarithmic equations to solve problems in computer science?

A: Logarithmic equations can be used to solve problems in computer science, such as analyzing algorithms and data structures. They can also be used to model computer systems and make predictions.

Q: What is the difference between a logarithmic equation and a recurrence relation?

A: A logarithmic equation and a recurrence relation are distinct concepts. A logarithmic equation is an equation that involves a logarithmic expression, such as $\log_2 8 = x$. A recurrence relation is an equation that involves a recursive formula, such as $a_n = 2a_{n-1}$.

Q: How do I use logarithmic equations to solve problems in mathematics?

A: Logarithmic equations can be used to solve problems in mathematics, such as analyzing functions and making predictions. They can also be used to model mathematical concepts, such as fractals and chaos theory.

Q: What is the significance of logarithmic equations in mathematics education?

A: Logarithmic equations have many applications in mathematics education, including teaching algebra and calculus. They can also be used to model mathematical concepts and make predictions.

Q: How do I use logarithmic equations to solve problems in statistics?

A: Logarithmic equations can be used to solve problems in statistics, such as analyzing data and making predictions. They can also be used to model statistical concepts, such as probability and hypothesis testing.

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

A: A logarithmic equation and a statistical equation are distinct concepts. A logarithmic equation is an equation that involves a logarithmic expression, such as $\log_2 8 = x$. A statistical equation is an equation that involves a statistical concept, such as probability or hypothesis testing.

Q: How do I use logarithmic equations to solve problems in physics?

A: Logarithmic equations can be used to solve problems in physics, such as analyzing motion and making predictions. They can also be used to model physical concepts, such as energy and momentum.

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