Math Problem Statement
Solve the equation 4^x = 21.
Solution
The equation presented in the image is:
To solve for , we can use logarithms.
Solution
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Take the natural logarithm (or common logarithm) of both sides:
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Use the power rule for logarithms, which states :
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Isolate by dividing both sides by :
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Calculate the value using approximate values for the natural logarithms:
Answer:
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Further Questions to Explore
- What is the effect of changing the base (4 in this case) on the solution of the equation?
- How would the solution change if the equation were ?
- Can we solve this equation using a different logarithm base, such as base 10?
- How do logarithmic functions relate to exponential functions like ?
- What are some real-life applications of solving exponential equations like this one?
Tip:
When solving exponential equations, using logarithms is often the most straightforward method to isolate the variable in the exponent.
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Math Problem Analysis
Mathematical Concepts
Algebra
Exponential Equations
Logarithms
Formulas
Power rule of logarithms: ln(a^b) = b * ln(a)
Change of base formula for logarithms
Theorems
Properties of logarithms
Suitable Grade Level
Grades 10-12