Question

Write the logarithm in terms of common logarithms.\(\ln 30\)

Short answer

The natural logarithm \( \ln 30 \) can be written in terms of common logarithms as approximately 3.4

Step-by-step solution

Write down the formula

The formula to convert a natural logarithm to a common logarithm is \( \ln a = \frac{\log_{10}a}{\log_{10}e} \). Write down this formula.

Plug in the value of 'a' into the formula

In this exercise, 'a' is 30. So, plug in this value into the formula: \( \ln 30 = \frac{\log_{10}30}{\log_{10}e} \).

Calculate the values \(\log_{10}30\) and \(\log_{10}e\)

The value of \(\log_{10}30\) is approximately 1.4771. In addition, since 'e' is approximately 2.71828, \(\log_{10}e\) is approximately 0.4343.

Compute the result

Now that you have computed the values of the numerator and the denominator of the fraction, divide them to get the result: \( \ln 30 = \frac{1.4771}{0.4343} \).

Simplify for the final answer

Dividing 1.4771 by 0.4343 will give you approximately 3.4. So, \( \ln 30 \approx 3.4 \).

Key concepts

Natural Logarithm

The natural logarithm is a concept commonly used in mathematics, especially in calculus and exponential growth problems. It is represented by the symbol \( \ln \), and it is the logarithm to the base \( e \). The number \( e \) is an irrational constant approximately equal to 2.71828.

Natural logarithms are instrumental when dealing with growth rates. This is because the base \( e \) is intimately connected with continuous growth processes, like compound interest or population growth.

• The notation \( \ln x \) indicates a logarithm at base \( e \).
• It is defined so that if \( y = \ln x \), then \( e^y = x \).
• It's widely used in mathematical models of real-world processes.

Understanding natural logarithms becomes essential when tackling problems involving exponential decay or growth, differential equations, and much more.

Common Logarithm

Common logarithms are logarithms with base 10. They are usually represented by \( \log_{10} \) or simply \( \log \) when the base is implied.

These logarithms are often used in science and engineering, because they simplify the process of dealing with very large or very small numbers, by converting multiplicative relationships into additive ones.

• Common logarithm of a number \( x \) is denoted as \( \log_{10}x \).
• They transform powers of 10 into integers, making it easier to calculate on paper.
• They were especially useful before the advent of calculators for simplifying complex computations.

Converting different forms of logarithms into common logarithms is a handy skill in disciplines that rely on numerical approximations and estimates.

Conversion Formula

The conversion formula is a useful tool for moving between different types of logarithms. This formula helps us express a natural logarithm in terms of a common logarithm, facilitating easier calculations, especially when calculators are limited to base 10 operations.

The formula states: \[ \ln a = \frac{\log_{10}a}{\log_{10}e} \]

Here's an example to understand it better:

• The \( \ln 30 \) can be converted using the formula above.
• By calculating \( \log_{10}30 \) to approximately 1.4771 and \( \log_{10}e \) to approximately 0.4343, we have \( \ln 30 = \frac{1.4771}{0.4343} \).
• This division leads to the approximate value of 3.4.

This conversion is instrumental when interchanging between different logarithmic bases in computational problems.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They are expressed in the form \( f(x) = \log_b(x) \), where \( b \) is the base. These functions play vital roles in many fields including statistics, acoustics, and even in solving certain equations in mathematics.

Some key points about logarithmic functions:

• They inverse the exponential functions, so if \( b^y = x \), \( y = \log_b(x) \).
• Logarithmic and exponential functions can model real-life phenomena like sound intensity, pH in chemistry, and Richter scale in seismology.
• Their graphs are smooth curves that increase or decrease steadily, which depends on the base.

Understanding the properties of logarithmic functions enables the solution of complex real-world problems involving growth or decay, showcasing their strong interdisciplinary nature.