Math Problem Statement

log 7 (15) to base e

Solution

To convert the logarithm log7(15)\log_7(15) to base ee, we use the change of base formula for logarithms:

logb(a)=ln(a)ln(b)\log_b(a) = \frac{\ln(a)}{\ln(b)}

Here, b=7b = 7 and a=15a = 15. So:

log7(15)=ln(15)ln(7)\log_7(15) = \frac{\ln(15)}{\ln(7)}

Now, using approximate values for the natural logarithms:

ln(15)2.7081andln(7)1.9459\ln(15) \approx 2.7081 \quad \text{and} \quad \ln(7) \approx 1.9459

log7(15)2.70811.94591.391\log_7(15) \approx \frac{2.7081}{1.9459} \approx 1.391

Thus, log7(15)\log_7(15) to base ee is approximately 1.391.

Would you like further details or have any questions?


Here are some related questions you might find interesting:

  1. What is the change of base formula for logarithms?
  2. How do you derive the natural logarithm (ln\ln) of a number?
  3. Can you calculate log7(20)\log_7(20) to base ee?
  4. What is the significance of natural logarithms in mathematics?
  5. How can logarithmic identities simplify complex logarithmic expressions?

Tip: The change of base formula is very useful when you need to compute logarithms with a calculator that only supports base 10 or base ee.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Change of Base Formula
Natural Logarithms

Formulas

log_b(a) = ln(a) / ln(b)

Theorems

Change of Base Theorem

Suitable Grade Level

Grades 10-12