Solve: Log2(x-1)+log2(x+5)=4 is a common logarithmic equation that students and math enthusiasts often encounter. Understanding how to approach and solve such equations is fundamental in algebra and helps build a strong foundation for more advanced mathematical concepts. In this article, we will explore the step-by-step process of solving the equation, clarify the key concepts involved, and provide tips to master similar problems.
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Understanding the Equation
Before diving into the solution, it is important to understand the structure of the equation:
Logarithmic equations involving sums of logs are often simplified using properties of logarithms. The given equation:
Log2(x-1) + log2(x+5) = 4
involves two logarithmic expressions with the same base, which suggests that we can combine them into a single logarithm.
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Key Concepts in Solving Logarithmic Equations
1. Logarithm Properties
To effectively solve the equation, recall the key properties of logarithms:- Product Property: logb(M) + logb(N) = log_b(M N)
- Change of Base: log_b(a) = ln(a) / ln(b), useful for conversions but not necessary here since base 2 is consistent
- Exponential Form: If log_b(x) = y, then x = b^y
In our case, the sum of logs can be combined into a single logarithm:
log2(x-1) + log2(x+5) = log2[(x-1)(x+5)]
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2. Domain Restrictions
Logarithmic functions are only defined for positive arguments:- x - 1 > 0 → x > 1
- x + 5 > 0 → x > -5
Since the more restrictive condition is x > 1, the domain of the solution must satisfy x > 1.
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Step-by-Step Solution
Step 1: Combine the Logarithms
Utilize the product property of logs:log2(x-1) + log2(x+5) = log2[(x-1)(x+5)]
So, the original equation becomes:
log2[(x-1)(x+5)] = 4
Step 2: Convert Logarithmic Equation to Exponential Form
Recall that if log_b(A) = C, then A = b^C. Applying this:(x-1)(x+5) = 2^4
Calculate 2^4:
(x-1)(x+5) = 16
Step 3: Expand and Form a Quadratic Equation
Expand the left side:x(x+5) - 1(x+5) = 16
x^2 + 5x - x - 5 = 16
x^2 + 4x - 5 = 16
Bring all to one side to set the quadratic to zero:
x^2 + 4x - 21 = 0
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Solving the Quadratic Equation
Step 4: Use the Quadratic Formula
The quadratic formula:x = [-b ± √(b^2 - 4ac)] / 2a
For the quadratic x^2 + 4x - 21 = 0:
a = 1, b = 4, c = -21
Calculate the discriminant:
D = b^2 - 4ac = 4^2 - 4(1)(-21) = 16 + 84 = 100
Find the roots:
x = [-4 ± √100] / 2 = [-4 ± 10] / 2
Calculate both solutions:
- x = (-4 + 10) / 2 = 6 / 2 = 3
- x = (-4 - 10) / 2 = -14 / 2 = -7
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Checking the Domain Restrictions
Recall that the original Logarithmic functions require x > 1.
- For x = 3: Since 3 > 1, it is within the domain.
- For x = -7: Since -7 is not > 1, it is outside the domain.
Therefore, the only valid solution is:
x = 3
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Final Answer and Verification
To verify, substitute x = 3 back into the original equation:
Log2(3-1) + log2(3+5) = ?
Calculate:
log2(2) + log2(8) = ?
Recall that:
log2(2) = 1 (since 2^1 = 2)
log2(8) = 3 (since 2^3 = 8)
Sum:
1 + 3 = 4
which matches the right side of the original equation.
Thus, x = 3 is the correct solution.
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Summary of the Solution Process
- Combine the logs using the product property.
- Convert the logarithmic equation into an exponential form.
- Form a quadratic equation and solve using the quadratic formula.
- Check the solutions against the domain restrictions.
- Verify the solutions by substitution.
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Additional Tips for Solving Logarithmic Equations
- Always analyze the domain restrictions before solving.
- Use logarithmic properties to simplify expressions.
- Convert logarithmic equations to exponential form for easier solving.
- Check all solutions in the original equation to avoid extraneous roots.
- Practice with different bases and more complex equations to strengthen your skills.
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Conclusion
The process of solving log2(x-1) + log2(x+5) = 4 illustrates fundamental principles of logarithms and algebra. By combining logs, converting to exponential form, solving the resulting quadratic, and verifying solutions, you can confidently tackle similar equations. Remember to always consider the domain restrictions imposed by the logarithmic functions to ensure your solutions are valid.
Mastering such techniques not only enhances your problem-solving skills but also prepares you for more advanced topics in mathematics, such as exponential functions, logarithmic inequalities, and calculus. Practice regularly with different types of logarithmic equations to become proficient and confident in solving them efficiently.
