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

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore how to solve a logarithmic equation involving a base of $\frac{3}{4}$ and an argument of $25$. We will break down the solution step by step, using mathematical concepts and formulas to arrive at the final answer.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, a logarithm answers the question "to what power must a base be raised to obtain a given number?" For example, the equation $\log_2 8 = 3$ asks "to what power must 2 be raised to obtain 8?" The answer, of course, is 3.

The Given Equation

The given equation is $\log_{\frac{3}{4}} 25 - 3x = -1$. This equation involves a logarithm with a base of $\frac{3}{4}$ and an argument of $25$. We are asked to find the approximate value of $x$.

Step 1: Isolate the Logarithmic Term

To solve the equation, we need to isolate the logarithmic term. We can do this by adding $3x$ to both sides of the equation:

$$\log_{\frac{3}{4}} 25 = -1 + 3x$$

Step 2: Use the Definition of a Logarithm

The definition of a logarithm states that $\log_b a = c$ is equivalent to $b^c = a$. We can use this definition to rewrite the equation:

$$(\frac{3}{4})^{-1} = 25$$

Step 3: Simplify the Equation

We can simplify the equation by evaluating the expression $(\frac{3}{4})^{-1}$:

$$\frac{4}{3} = 25$$

This equation is not true, so we need to go back to the original equation and try a different approach.

Step 4: Use the Change of Base Formula

The change of base formula states that $\log_b a = \frac{\log_c a}{\log_c b}$ for any positive numbers $a$, $b$, and $c$. We can use this formula to rewrite the equation:

$$\frac{\log 25}{\log \frac{3}{4}} = -1 + 3x$$

Step 5: Evaluate the Logarithms

We can evaluate the logarithms using a calculator or a logarithmic table:

$$\log 25 \approx 1.39794$$

$$\log \frac{3}{4} \approx -0.12532$$

Step 6: Simplify the Equation

We can simplify the equation by substituting the values of the logarithms:

$$\frac{1.39794}{-0.12532} = -1 + 3x$$

Step 7: Solve for $x$

We can solve for $x$ by isolating the variable:

$$-11.176 = -1 + 3x$$

$$-10.176 = 3x$$

$$x \approx -3.392$$

Conclusion

In this article, we solved a logarithmic equation involving a base of $\frac{3}{4}$ and an argument of $25$. We used the definition of a logarithm, the change of base formula, and logarithmic properties to arrive at the final answer. The approximate value of $x$ is $-3.392$.

Answer

The correct answer is:

• A. -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. In other words, a logarithm answers the question "to what power must a base be raised to obtain a given number?"

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 definition of a logarithm to rewrite the equation. You can also use the change of base formula to rewrite the equation in a more manageable form.

Q: What is the change of base formula?

A: The change of base formula states that $\log_b a = \frac{\log_c a}{\log_c b}$ for any positive numbers $a$, $b$, and $c$. This formula allows you to rewrite a logarithmic equation in terms of a different base.

Q: How do I evaluate logarithms?

A: You can evaluate logarithms using a calculator or a logarithmic table. You can also use the properties of logarithms, such as the product rule and the quotient rule, to simplify the logarithmic expression.

Q: What are some common logarithmic properties?

A: Some common logarithmic properties include:

• The product rule: $\log_b (xy) = \log_b x + \log_b y$
• The quotient rule: $\log_b \frac{x}{y} = \log_b x - \log_b y$
• The power rule: $\log_b x^y = y \log_b x$

Q: How do I use the definition of a logarithm to solve an equation?

A: To use the definition of a logarithm to solve an equation, you need to rewrite the equation in the form $b^c = a$, where $b$ is the base, $c$ is the exponent, and $a$ is the argument. You can then solve for the exponent $c$.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, the equation $\log_2 8 = 3$ is a logarithmic equation, while the equation $2^3 = 8$ is an exponential equation.

Q: Can you give an example of a logarithmic equation?

A: Yes, here is an example of a logarithmic equation:

$$\log_2 8 - 3x = -1$$

This equation involves a logarithm with a base of 2 and an argument of 8. We can solve for $x$ by isolating the logarithmic term and then using the definition of a logarithm to rewrite the equation.

Q: Can you give an example of a logarithmic equation with a base of $\frac{3}{4}$?

A: Yes, here is an example of a logarithmic equation with a base of $\frac{3}{4}$:

$$\log_{\frac{3}{4}} 25 - 3x = -1$$

This equation involves a logarithm with a base of $\frac{3}{4}$ and an argument of 25. We can solve for $x$ by isolating the logarithmic term and then using the definition of a logarithm to rewrite the equation.

Conclusion

In this article, we answered some frequently asked questions about logarithmic equations. We discussed the definition of a logarithmic equation, how to solve a logarithmic equation, and some common logarithmic properties. We also gave examples of logarithmic equations with different bases.