Math Problem Statement
Find the exact value of each expression: (a) ln(e^7), (b) e^(ln(4)), (c) e^(ln(√4)), (d) ln(1/e^4).
Solution
Let's solve the given logarithmic and exponential expressions from the homework.
(a)
Using the property of logarithms, , we have:
(b)
Using the inverse property , we have:
(c)
First simplify , so the expression becomes:
(d)
We can rewrite this expression as: Using the logarithmic property , we have:
Summary of answers:
- (a)
- (b)
- (c)
- (d)
Would you like a deeper explanation of any of these steps?
Here are 5 related questions to consider:
- How do logarithmic and exponential functions relate to each other?
- What are some common properties of logarithms that simplify calculations?
- Why does hold for any real number ?
- What is the domain of the natural logarithm function?
- How can we apply logarithmic properties to solve real-world problems?
Tip: Always look for properties like and to simplify expressions quickly!
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Math Problem Analysis
Mathematical Concepts
Logarithms
Exponential Functions
Natural Logarithms
Formulas
ln(e^x) = x
e^(ln(x)) = x
ln(1/e^x) = -x
Theorems
Properties of Logarithms
Inverse Relationship Between Exponentials and Logarithms
Suitable Grade Level
High School (Grades 9-12)