Math Problem Statement
Lab Worksheet 0 (2).pdf
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solve question 3
Solution
The given equations in question 3 are:
where and are constants. We need to express as a function of .
Step 1: Express in Terms of Other Variables
Taking the natural logarithm on both sides of the first equation:
Using logarithm properties:
Rearrange for :
Step 2: Express in Terms of Other Variables
From the second equation:
Solving for :
Step 3: Substitute into the Expression for
Substituting into the equation for :
Expanding the logarithm:
Step 4: Solve for
Rearrange to isolate :
Factor out:
Solving for :
Thus, the function for in terms of is:
Would you like a detailed numerical example to illustrate this?
Related Questions:
- How would the solution change if and ?
- What happens to if ?
- How can logarithmic identities be used to simplify this result further?
- What happens if in the given equations?
- Can this equation be solved graphically for as a function of ?
Tip:
When solving logarithmic equations, always check for domain restrictions to ensure valid solutions!
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Math Problem Analysis
Mathematical Concepts
Logarithms
Exponential Equations
Algebraic Manipulation
Formulas
ln(ap^q) = ln(bt^r)
ln a + q ln p = ln b + r ln t
b^s q = at
Theorems
Logarithmic Properties
Exponentiation Rules
Suitable Grade Level
Undergraduate Mathematics / Advanced High School