Math Problem Statement
^{\placeholder{}} $$$$ 3^{x+1}=\exponentialE^7
Solution
You want to solve the equation .
Step-by-Step Solution:
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Start with the equation:
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Apply the natural logarithm (ln) to both sides to simplify the exponents:
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Use the logarithmic identity to simplify:
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Solve for :
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Finally, solve for by subtracting 1:
This is the solution for .
Would you like a detailed explanation or clarification on any step?
Here are 5 related questions to consider:
- How do logarithmic identities like work in different contexts?
- What is the significance of the natural logarithm in solving exponential equations?
- How does base (Euler's number) differ from other bases like 3 or 10 in logarithmic functions?
- How can you graph functions like and to visually find the solution for ?
- What methods exist to approximate solutions when logarithmic expressions are involved?
Tip: When solving exponential equations, applying logarithms can simplify exponents into linear equations, making them much easier to handle.
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Math Problem Analysis
Mathematical Concepts
Algebra
Exponential Equations
Logarithms
Formulas
Natural logarithm: ln(a^b) = b * ln(a)
Exponential equation: 3^{x+1} = e^7
Solution for x in exponential equations: x = (7 / ln(3)) - 1
Theorems
Logarithmic identity: ln(a^b) = b * ln(a)
Properties of logarithms: ln(e^x) = x
Suitable Grade Level
Grades 10-12