Math Problem Statement
<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-cke-white-space-first="1" data-cke-white-space-last="1"> <mo>log</mo> <mfenced close=")" open="("> <mrow> <mi mathvariant="italic">x</mi> </mrow> </mfenced> <mo>−</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>log</mo> <mfenced close=")" open="("> <mrow> <mi mathvariant="italic">y</mi> </mrow> </mfenced> <mo>+</mo> <mn>7</mn> <mo>log</mo> <mfenced close=")" open="("> <mrow> <mi mathvariant="italic">z</mi> </mrow> </mfenced><mo>=</mo> </mrow></math>
Solution
We are tasked with simplifying or solving the logarithmic equation:
Step-by-step explanation:
-
Apply the logarithmic property :
- For , we rewrite it as:
- For , we rewrite it as:
Substituting these into the equation, we get:
-
Combine the logarithms using the property :
- Combine all terms into a single logarithm:
Simplify further:
-
Exponentiate to remove the logarithm:
Recall that . Therefore:
Since , we have:
-
Solve for :
Multiply through by :
Finally:
Final Simplified Expression:
Would you like further clarification or examples on logarithmic properties?
Related Questions:
- What are the laws of logarithms and how are they applied in equations?
- How do you handle equations involving logarithms with different bases?
- What is the geometric interpretation of logarithmic equations like this one?
- How can we extend this method to solve systems of logarithmic equations?
- What are common mistakes to avoid when simplifying logarithmic expressions?
Tip:
When solving logarithmic equations, always check for domain restrictions to ensure all variables meet the conditions , , and .
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Math Problem Analysis
Mathematical Concepts
Logarithmic Equations
Properties of Logarithms
Simplification of Expressions
Formulas
a * log(b) = log(b^a)
log(a) + log(b) = log(a * b)
log(a) = b implies a = 10^b
Theorems
Properties of Logarithms
Exponentiation to Remove Logarithms
Suitable Grade Level
Grades 10-12