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Chapter 5: Problem 77

Find the logarithm using natural logarithms and the change-of-base formula. $$\log _{100} 15$$

Short Answer

Expert verified
Using the change-of-base formula with natural logarithms, \( log_{100} 15 \approx 0.5888 \).

Step by step solution

01

Understand the Change-of-Base Formula

The change-of-base formula states that for any positive numbers a, b, and base c (where c 鈮� 1): \ \ \ \ \ \( \ log_b a = \ \frac \ {log_c a} \ {log_c b} \)
02

Choose the Base for Logarithms

In this case, use natural logarithms (base e). This means we will use the formula: \ \ \( \ log_{100} 15 = \ \frac {ln 15} \ {ln 100} \)
03

Calculate the Natural Logarithms

Compute \( \ ln 15 \) and \( \ ln 100 \). \ \ Using a calculator, \ \ \( \ ln 15 \approx \ 2.70805 \ \) \ \ and \ \( ln 100 = 4.60517 \).
04

Apply the Change-of-Base Formula

Now plug these values into the formula: \ \ \( log_{100} 15 = \ \frac {ln 15} \ {ln 100} \approx \ \frac {2.70805} \ {4.60517} \).
05

Simplify the Fraction

Simplify \( \ \ \frac {2.70805} \ {4.60517} \). This gives: \ \ \( \ \ log_{100} 15 \approx 0.5888 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithms
Natural logarithms are logarithms with base e, where e is approximately 2.71828. These logarithms are denoted as ln(x). Natural logarithms are frequently used in various mathematical calculations due to their unique properties. For instance, they arise naturally in problems involving growth or decay, such as population growth or radioactive decay.
Natural logarithms are useful when dealing with exponential functions and simplifying complex logarithmic expressions.
In the given problem, we use natural logarithms to apply the change-of-base formula.
Base Change
Changing the base of a logarithm allows us to simplify computations by converting logarithms to a more convenient base. The change-of-base formula is used to transform a logarithm of any base to a logarithm of a chosen base. This formula is:
\( \log_b a = \frac{\log_c a}{\log_c b} \)
Here, a and b are any positive numbers, and c is the new base (c 鈮� 1).
In this problem, we chose the natural logarithm (base e) to simplify our calculations:
\( \log_{100} 15 = \frac{\ln 15}{\ln 100} \).
Logarithmic Calculations
Logarithmic calculations involve finding the logarithm of a number, which is the exponent to which the base must be raised to yield that number. In the given problem, we needed to find \( \log_{100} 15 \). Using natural logarithms and the change-of-base formula, we can convert this to a more manageable form.
First, we find the natural logarithms:\( \ ln 15 \approx 2.70805 \) and \( \ ln 100 = 4.60517 \).

Next, we use the change-of-base formula:
\( \log_{100} 15 = \frac{\ln 15}{\ln 100} \approx \frac{2.70805}{4.60517} \approx 0.5888 \).

This final value is the logarithm of 15 with base 100, simplified with natural logarithms.

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Most popular questions from this chapter

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