Question

Change of Base Find the common logarithm of the number whose natural logarithm is the given value. $$15.36$$

Short answer

The common logarithm of the number whose natural logarithm is 15.36 is approximately 6.674.

Step-by-step solution

Understanding the Problem

We are given the natural logarithm (ln) of a number, which is 15.36. We are asked to find the common logarithm (log base 10) of the same number. To do this, we can use the change of base formula for logarithms.

Change of Base Formula

The change of base formula allows us to convert between logarithms of different bases. It is given by: \[ \log_b a = \frac{\log_c a}{\log_c b} \] In our case, we want to find the common logarithm (base 10) of a number given the natural logarithm (base e), which can be represented as: \[ \log a = \frac{\ln a}{\ln 10} \]

Applying the Change of Base Formula

Using the value given for the natural logarithm of the number, we substitute into the change of base formula: \[ \log a = \frac{15.36}{\ln 10} \] Next, we need to calculate the value of ln(10).

Calculating ln(10)

Using a calculator or known value, \( \ln 10 \) is approximately 2.3026. Substituting this value into the formula gives us: \[ \log a = \frac{15.36}{2.3026} \]

Final Calculation

Performing the division yields the common logarithm (base 10) of the number: \[ \log a \approx \frac{15.36}{2.3026} \approx 6.674 \] This is the common logarithm of the number whose natural logarithm is 15.36.

Key concepts

Common Logarithm

When we talk about the common logarithm, we're referring to a logarithmic function with base 10. This is the logarithm we use when no base is specified, and is typically denoted as \( \text{log} \(a\) \) without any subscript. It is a fundamental concept in mathematics because our number system is based on the base 10, making common logarithms particularly intuitive for representing the order of magnitude of numbers.

For example, if you want to calculate the common logarithm of 100, you would write \( \text{log} \(100\) \) which equals 2. This means that the power to which the base 10 must be raised to get 100 is 2, since \( 10^2 = 100 \). In the context of the provided exercise where the natural logarithm of a number is known, converting to a common logarithm involves using a mathematical procedure known as logarithmic conversion.

This conversion utilizes the change of base formula and is crucial in various fields, including engineering, sciences, and finance, where the comparison of large numbers is frequently required. Understanding the common logarithm serves as a stepping stone for higher mathematical concepts and practical applications alike.

Natural Logarithm

The natural logarithm is another type of logarithmic function, often denoted as \( \text{ln} \(x\) \), with the special number \( e \) (approximately 2.71828) as its base. The natural logarithm is widely used in mathematics, particularly in calculus, because it has properties that make it convenient for describing growth processes and calculating areas under curves, among other things.

When you see \( \text{ln} \(x\) \), it represents the power to which you need to raise \( e \) to get \( x \). So, if we have \( \text{ln} \(7.389\) \), it is approximately 2 because \( e^2 \) is roughly equal to 7.389. The exercise presents a scenario where you are given the value of a natural logarithm and are required to translate it into a common logarithm, providing a practical application of the natural logarithm in the context of logarithmic conversion.

Logarithmic Conversion

The process of changing from one logarithm base to another is known as logarithmic conversion. It is an important skill because different situations require different bases. For example, in computer science, the base 2 logarithm is predominant due to binary computation, whereas common logarithms are used in pH calculations in chemistry.

The change of base formula underpins this conversion process, enabling us to convert a logarithm from one base to another with ease. In the context of the exercise, knowing the natural logarithm of a number, we use the formula to find its common logarithm. This formula is expressed as \[ \text{log}_b a = \frac{\text{log}_c a}{\text{log}_c b} \], where \( b \) and \( c \) are the bases you're converting between, and \( a \) is the number whose logarithm you're calculating.

• The numerator is the logarithm of \( a \) with respect to the new base \( c \).
• The denominator is the logarithm of the old base \( b \) with respect to the new base \( c \).

Using the change of base formula doesn't just provide us with a number. It also enhances our understanding of the relative magnitude of logarithms across different bases and solidifies our comprehension of exponential and logarithmic relationships in mathematics.