Math Problem Statement
Solution
Let's go through each equation and solve for .
1)
Express both sides with base 2:
Thus, the equation becomes:
Simplify:
Equating exponents:
Solve for :
2)
Rewrite as :
Simplify:
Equating exponents:
Solve for :
3)
Express both sides with base 2:
Rewrite the equation:
Simplify:
Equating exponents:
Solve for :
4)
Rewrite as :
Taking logarithms (or using exponents properties):
Solve for :
5)
Take the logarithm of both sides:
Note that . The equation becomes:
Solve for :
Would you like me to continue with the rest of the questions, or do you need further clarification on these solutions? Here are some related questions you might be interested in:
- How do logarithmic properties help solve exponential equations?
- Can all exponential equations be solved by converting to the same base?
- What are the general steps to solving inequalities involving exponents?
- How do you handle fractional bases in exponential equations?
- Why do exponential functions often appear in growth or decay models?
Tip: When solving equations involving exponents, converting both sides to the same base often simplifies the process significantly!
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Math Problem Analysis
Mathematical Concepts
Exponential Equations
Algebra
Logarithms
Inequalities
Formulas
a^x = b^y → x = y (if a = b)
Logarithmic properties
Properties of exponents
Theorems
Laws of Exponents
Logarithmic identities
Solving exponential equations by equating exponents
Suitable Grade Level
Grades 9-12
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