Solve The Equation: Log ⁡ ( 3 X ) + Log ⁡ ( 2 X − 1 ) = Log ⁡ ( 16 X − 10 \log (3x) + \log (2x - 1) = \log (16x - 10 Lo G ( 3 X ) + Lo G ( 2 X − 1 ) = Lo G ( 16 X − 10 ]

Mar 10, 2025 by ADMIN 170 views

**Solving Logarithmic Equations: A Step-by-Step Guide**

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Introduction

Logarithmic equations can be challenging to solve, especially when they involve multiple logarithmic terms. In this article, we will focus on solving a specific logarithmic equation that involves the sum of two logarithmic terms. The equation is given as:

\log (3x) + \log (2x - 1) = \log (16x - 10)

Understanding Logarithmic Properties

Before we dive into solving the equation, it's essential to understand the properties of logarithms. The logarithmic function is the inverse of the exponential function. The logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number.

One of the key properties of logarithms is the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual terms. Mathematically, this can be expressed as:

\log (ab) = \log a + \log b

Another important property is the quotient rule, which states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Mathematically, this can be expressed as:

\log \left(\frac{a}{b}\right) = \log a - \log b

Solving the Logarithmic Equation

Now that we have a good understanding of logarithmic properties, let's focus on solving the given equation. The equation is:

\log (3x) + \log (2x - 1) = \log (16x - 10)

Using the product rule, we can rewrite the left-hand side of the equation as:

\log (3x(2x - 1)) = \log (16x - 10)

Evaluating the Logarithmic Terms

The next step is to evaluate the logarithmic terms on both sides of the equation. Since the logarithmic function is the inverse of the exponential function, we can rewrite the equation in exponential form as:

3x(2x - 1) = 16x - 10

Simplifying the Equation

Now that we have the equation in exponential form, let's simplify it by expanding the left-hand side:

6x^2 - 3x = 16x - 10

Rearranging the Equation

The next step is to rearrange the equation to get all the terms on one side:

6x^2 - 3x - 16x + 10 = 0

Combining Like Terms

Now that we have the equation in a more manageable form, let's combine like terms:

6x^2 - 19x + 10 = 0

Solving the Quadratic Equation

The equation we have is a quadratic equation, which can be solved using various methods such as factoring, completing the square, or using the quadratic formula. In this case, we will use the quadratic formula to solve the equation.

The quadratic formula is given by:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

Applying the Quadratic Formula

Now that we have the quadratic formula, let's apply it to our equation:

x = \frac{-(-19) \pm \sqrt{(-19)^2 - 4(6)(10)}}{2(6)}

Simplifying the Expression

The next step is to simplify the expression under the square root:

x = \frac{19 \pm \sqrt{361 - 240}}{12}

Evaluating the Square Root

The next step is to evaluate the square root:

x = \frac{19 \pm \sqrt{121}}{12}

Simplifying the Expression

The next step is to simplify the expression:

x = \frac{19 \pm 11}{12}

Finding the Solutions

The final step is to find the solutions to the equation. We have two possible solutions:

x = \frac{19 + 11}{12} \quad \text{or} \quad x = \frac{19 - 11}{12}

Simplifying the Solutions

The next step is to simplify the solutions:

x = \frac{30}{12} \quad \text{or} \quad x = \frac{8}{12}

Reducing the Fractions

The final step is to reduce the fractions:

x = \frac{5}{2} \quad \text{or} \quad x = \frac{2}{3}

Conclusion

In this article, we have solved a logarithmic equation that involved the sum of two logarithmic terms. We used the product rule and the quotient rule to simplify the equation and then applied the quadratic formula to find the solutions. The final solutions were $x = \frac{5}{2}$ and $x = \frac{2}{3}$ .

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Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function. The logarithmic function is the inverse of the exponential function, and it is used to solve equations that involve exponential expressions.

Q: What are the common properties of logarithms?

A: The common properties of logarithms include the product rule, the quotient rule, and the power rule. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual terms. The quotient rule states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. The power rule states that the logarithm of a power is equal to the exponent times 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 to simplify the equation and then apply the quadratic formula to find the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is given by:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

Q: How do I apply the quadratic formula to a logarithmic equation?

A: To apply the quadratic formula to a logarithmic equation, you need to first simplify the equation using the properties of logarithms and then plug the values of $a$ , $b$ , and $c$ into the quadratic formula.

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

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

Not using the properties of logarithms to simplify the equation
Not applying the quadratic formula correctly
Not checking the solutions to see if they are valid
Not considering the domain of the logarithmic function

Q: How do I check the solutions to a logarithmic equation?

A: To check the solutions to a logarithmic equation, you need to plug the solutions back into the original equation and check if they are true. You also need to check if the solutions are valid, meaning that they are in the domain of the logarithmic function.

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

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

Modeling population growth and decay
Modeling chemical reactions
Modeling financial transactions
Modeling physical phenomena such as sound waves and light waves

Q: How do I use technology to solve logarithmic equations?

A: There are many software programs and online tools that can be used to solve logarithmic equations, including:

Graphing calculators
Computer algebra systems
Online equation solvers
Spreadsheets

Q: What are some tips for solving logarithmic equations?

A: Some tips for solving logarithmic equations include:

Using the properties of logarithms to simplify the equation
Applying the quadratic formula correctly
Checking the solutions to see if they are valid
Considering the domain of the logarithmic function
Using technology to check the solutions and simplify the equation

Conclusion

In this article, we have answered some frequently asked questions about solving logarithmic equations. We have covered topics such as the properties of logarithms, the quadratic formula, and common mistakes to avoid. We have also discussed real-world applications of logarithmic equations and provided tips for solving them.