Solving Logarithmic Equations A Comprehensive Guide

This article provides a comprehensive guide to solving logarithmic equations. We will explore various techniques and strategies to tackle different types of logarithmic equations, including those with different bases and complex expressions. Our focus will be on understanding the fundamental properties of logarithms and applying them to simplify and solve equations. We will delve into examples that involve changing bases, using the properties of logarithms to combine terms, and dealing with extraneous solutions. By mastering these techniques, you will be well-equipped to solve a wide range of logarithmic equations.

(a) Solving the Equation: log₃ x = log₉ (x+12)

In this section, we will tackle the logarithmic equation log₃ x = log₉ (x+12). This equation presents a classic scenario where the logarithms have different bases, namely 3 and 9. To effectively solve this, our primary strategy will be to unify the bases. The key observation here is that 9 is a power of 3 (9 = 3²), which allows us to express the logarithm with base 9 in terms of a logarithm with base 3. This transformation is crucial for simplifying the equation and making it solvable. The change of base formula is a fundamental tool in this process, allowing us to rewrite logarithms in different bases. Once we have a common base, we can then use the properties of logarithms to combine terms and eliminate the logarithms, leading to a simpler algebraic equation. This equation can then be solved using standard algebraic techniques. However, it's essential to remember that when solving logarithmic equations, we must always check for extraneous solutions. These are solutions that satisfy the algebraic equation but do not satisfy the original logarithmic equation due to domain restrictions (logarithms are only defined for positive arguments). Therefore, after finding potential solutions, we will carefully substitute them back into the original equation to ensure their validity. This step is a crucial part of solving logarithmic equations accurately. In this specific equation, we will encounter a quadratic equation after simplification. Solving the quadratic equation will give us two potential solutions. However, only one of these solutions will satisfy the domain restrictions of the logarithmic functions in the original equation. The other solution will be an extraneous solution and must be discarded. This highlights the importance of checking for extraneous solutions in logarithmic equations.

Step-by-step Solution

1. Change the base of log₉ (x+12) to base 3: Using the change of base formula, we have: log₉ (x+12) = log₃ (x+12) / log₃ 9 = log₃ (x+12) / 2
2. Rewrite the original equation: The equation log₃ x = log₉ (x+12) becomes: log₃ x = (1/2) log₃ (x+12)
3. Multiply both sides by 2: 2 log₃ x = log₃ (x+12)
4. Use the power rule of logarithms: log₃ (x²) = log₃ (x+12)
5. Since the bases are the same, equate the arguments: x² = x + 12
6. Rearrange into a quadratic equation: x² - x - 12 = 0
7. Factor the quadratic equation: (x - 4)(x + 3) = 0
8. Solve for x: x = 4 or x = -3
9. Check for extraneous solutions:
   • For x = 4: log₃ 4 and log₉ (4+12) = log₉ 16 are both defined.
   • For x = -3: log₃ (-3) is undefined, so x = -3 is an extraneous solution.

Therefore, the only valid solution is x = 4.

(b) Solving the Equation: 4logₓ 2 - log₂ x = 3

Now, let's tackle the logarithmic equation 4logₓ 2 - log₂ x = 3. This equation presents a slightly different challenge compared to the previous one. Here, we have logarithms with bases x and 2. The key to solving this equation lies in recognizing the reciprocal relationship between logₓ 2 and log₂ x. Specifically, we know that logₓ 2 = 1 / log₂ x. This relationship is a direct consequence of the change of base formula and is a powerful tool for simplifying logarithmic expressions and equations. By substituting this reciprocal relationship into the original equation, we can transform it into a more manageable form. This substitution will allow us to express the entire equation in terms of a single logarithm, log₂ x. Once we have the equation in this form, we can employ algebraic techniques to solve for log₂ x. This often involves making a substitution, such as y = log₂ x, to transform the equation into a polynomial equation, which we can then solve using standard methods like factoring or the quadratic formula. After finding the values of log₂ x, we can then solve for x by using the definition of logarithms. However, as with all logarithmic equations, we must remember to check for extraneous solutions. The domain of a logarithmic function requires the argument to be positive and the base to be positive and not equal to 1. Therefore, we need to verify that the solutions we obtain for x satisfy these conditions. Any solution that violates these conditions is an extraneous solution and must be discarded. In this particular equation, we will find that substituting the reciprocal relationship leads to a quadratic equation in terms of log₂ x. Solving this quadratic equation will give us two possible values for log₂ x, which in turn will give us two possible values for x. We will then need to check each of these solutions to ensure they satisfy the domain restrictions of the original logarithmic equation.

Step-by-step Solution

1. Use the change of base formula to rewrite logₓ 2: logₓ 2 = 1 / log₂ x
2. Substitute this into the original equation: 4(1 / log₂ x) - log₂ x = 3
3. Let y = log₂ x: 4/y - y = 3
4. Multiply both sides by y: 4 - y² = 3y
5. Rearrange into a quadratic equation: y² + 3y - 4 = 0
6. Factor the quadratic equation: (y + 4)(y - 1) = 0
7. Solve for y: y = -4 or y = 1
8. Substitute back log₂ x for y:
   • log₂ x = -4 => x = 2⁻⁴ = 1/16
   • log₂ x = 1 => x = 2¹ = 2
9. Check for extraneous solutions:
   • For x = 1/16: 4log_(1/16) 2 - log₂ (1/16) = 4(-1/4) - (-4) = -1 + 4 = 3. This solution is valid.
   • For x = 2: 4log₂ 2 - log₂ 2 = 4(1) - 1 = 3. This solution is also valid.

Therefore, the solutions are x = 1/16 and x = 2.

(c) Solving the Equation: 2logₓ 3 - log₃ √x = 3/2

Let's move on to the logarithmic equation 2logₓ 3 - log₃ √x = 3/2. This equation involves a combination of logarithmic terms with different bases (x and 3) and a square root within a logarithm. To solve this equation effectively, we will need to employ several key strategies. First, we will utilize the change of base formula to express all logarithmic terms in a common base. This will allow us to combine the terms more easily and simplify the equation. Similar to the previous example, we will use the reciprocal relationship between logarithms with interchanged bases. Specifically, we will use the fact that logₓ 3 = 1 / log₃ x. This substitution will help us to unify the logarithmic terms in the equation. Second, we will simplify the term log₃ √x. Recall that √x can be written as x^(1/2). Using the power rule of logarithms, we can rewrite log₃ √x as (1/2)log₃ x. This simplification will make the equation easier to manipulate. After making these substitutions and simplifications, we will likely obtain an equation that can be further simplified by making a substitution. For instance, we might let y = log₃ x. This will transform the equation into a more familiar algebraic form, such as a quadratic equation. We can then solve this algebraic equation for y using standard techniques. Once we have the values of y, we can substitute back log₃ x for y and solve for x. As always, it is crucial to check for extraneous solutions. We need to ensure that the values we obtain for x satisfy the domain restrictions of the original logarithmic equation. This means that x must be positive and not equal to 1, and any expression inside a logarithm must also be positive. By carefully checking for extraneous solutions, we can ensure that we have found all the valid solutions to the logarithmic equation.

Step-by-step Solution

1. Use the change of base formula to rewrite logₓ 3: logₓ 3 = 1 / log₃ x
2. Rewrite log₃ √x using the power rule: log₃ √x = log₃ (x^(1/2)) = (1/2) log₃ x
3. Substitute these into the original equation: 2(1 / log₃ x) - (1/2) log₃ x = 3/2
4. Let y = log₃ x: 2/y - (1/2)y = 3/2
5. Multiply both sides by 2y: 4 - y² = 3y
6. Rearrange into a quadratic equation: y² + 3y - 4 = 0
7. Factor the quadratic equation: (y + 4)(y - 1) = 0
8. Solve for y: y = -4 or y = 1
9. Substitute back log₃ x for y:
   • log₃ x = -4 => x = 3⁻⁴ = 1/81
   • log₃ x = 1 => x = 3¹ = 3
10. Check for extraneous solutions:
    • For x = 1/81: 2log_(1/81) 3 - log₃ √(1/81) = 2(-1/4) - (-4/2) = -1/2 + 2 = 3/2. This solution is valid.
    • For x = 3: 2log₃ 3 - log₃ √3 = 2(1) - 1/2 = 3/2. This solution is also valid.

Therefore, the solutions are x = 1/81 and x = 3.

Conclusion

In conclusion, solving logarithmic equations requires a strong understanding of the properties of logarithms and careful application of algebraic techniques. We've seen how the change of base formula, the power rule, and the reciprocal relationship between logarithms can be used to simplify equations and transform them into solvable forms. It is also crucial to remember to check for extraneous solutions, as logarithmic functions have domain restrictions that can lead to invalid solutions. By mastering these techniques, you can confidently solve a wide variety of logarithmic equations.