Math Problem Statement
Log (x - 1) + log (x + 1) = 1, find the value of x.
Solution
We are given the equation:
Step 1: Use the logarithmic property
We can combine the logarithms using the property of logarithms:
So, applying this to the given equation:
Step 2: Simplify the expression
The product simplifies to:
Now, we have:
Step 3: Convert from logarithmic form to exponential form
Recall that implies . Here, since we are using the common logarithm (base 10), we convert the equation to:
Step 4: Solve for
Now, solve the equation:
Step 5: Check for validity
Since the logarithmic function is defined only for positive arguments, we must ensure that both and . This implies:
Therefore, is the only valid solution (as would make undefined).
Final Answer:
The value of is:
Would you like more details on any step?
Here are 5 related questions to practice:
- Solve .
- Solve .
- If , find .
- Solve .
- Find if .
Tip: Always check the domain restrictions when solving logarithmic equations, as logarithms are only defined for positive values.
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Math Problem Analysis
Mathematical Concepts
Logarithms
Algebra
Exponential Equations
Formulas
log(a) + log(b) = log(a * b)
log_b(y) = z implies y = b^z
Theorems
Logarithmic properties
Exponential form of logarithmic equations
Suitable Grade Level
Grades 9-12