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) ln(e7)\ln(e^7)

Using the property of logarithms, ln(ex)=x\ln(e^x) = x, we have: ln(e7)=7\ln(e^7) = 7

(b) eln(4)e^{\ln(4)}

Using the inverse property eln(x)=xe^{\ln(x)} = x, we have: eln(4)=4e^{\ln(4)} = 4

(c) eln(4)e^{\ln(\sqrt{4})}

First simplify 4=2\sqrt{4} = 2, so the expression becomes: eln(2)=2e^{\ln(2)} = 2

(d) ln(1e4)\ln\left(\frac{1}{e^4}\right)

We can rewrite this expression as: ln(e4)\ln\left(e^{-4}\right) Using the logarithmic property ln(ex)=x\ln(e^x) = x, we have: ln(e4)=4\ln\left(e^{-4}\right) = -4

Summary of answers:

  • (a) ln(e7)=7\ln(e^7) = 7
  • (b) eln(4)=4e^{\ln(4)} = 4
  • (c) eln(4)=2e^{\ln(\sqrt{4})} = 2
  • (d) ln(1e4)=4\ln\left(\frac{1}{e^4}\right) = -4

Would you like a deeper explanation of any of these steps?

Here are 5 related questions to consider:

  1. How do logarithmic and exponential functions relate to each other?
  2. What are some common properties of logarithms that simplify calculations?
  3. Why does ln(ex)=x\ln(e^x) = x hold for any real number xx?
  4. What is the domain of the natural logarithm function?
  5. How can we apply logarithmic properties to solve real-world problems?

Tip: Always look for properties like ln(ex)=x\ln(e^x) = x and eln(x)=xe^{\ln(x)} = x 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)