Math Problem Statement
Solution
The problem in the image is asking for the value of in terms of , given the equation:
Let's start by breaking it down and solving step by step.
Step 1: Express as a product of primes
Since , we can rewrite the equation as:
This simplifies to:
Step 2: Compare the powers of 2 on both sides
On the left-hand side, we have , and on the right-hand side, the factor involving is . To satisfy this equation, the powers of 2 must be equal. Therefore, we equate the powers of 2:
Simplifying this equation:
Step 3: Find
We are asked to find in terms of . Since , we substitute this back into the original equation:
Dividing both sides by gives:
Thus, we find:
Final Answer:
The value of is when .
Would you like further clarification or additional steps explained?
Here are five questions to expand on this concept:
- How do you solve exponential equations when variables are in the exponent?
- What are some common techniques for dealing with exponential equations involving different bases?
- Can you generalize this problem for any constant besides 14?
- How does logarithmic manipulation help in solving exponential equations?
- How can we apply this concept to real-world growth or decay problems?
Tip: When solving exponential equations, always look for ways to express numbers as powers of the same base to simplify comparisons.
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Math Problem Analysis
Mathematical Concepts
Exponential Equations
Prime Factorization
Algebra
Formulas
Prime factorization: 14 = 2 × 7
Exponent comparison
Solving for variables
Theorems
Laws of Exponents
Exponentiation and base manipulation
Suitable Grade Level
Grades 10-12