Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Write the logarithm in terms of natural logarithms.\(\log _{7} 12\)

Step-by-Step Solution

Answer

The logarithm \( \log _{7} 12 \) in terms of natural logarithms is \( \frac{\ln 12}{\ln 7} \)

Step 1: Identify the given values

The given values in this exercise are the base (7) and the argument (12) in the logarithmic function, which are the values that will be used in the change of base formula. So, \(a = 12\) and \(b = 7\).

Step 2: Apply the change of base formula

The change of base formula, \(\log _{b}a = \frac{\ln a}{\ln b}\), can be applied to the original logarithmic function. By substituting \(a = 12\) and \(b = 7\), this formula can be rewritten as \(\log _{7}12 = \frac{\ln 12}{\ln 7}\).

Step 3: Simplify the expression

The expression \(\frac{\ln 12}{\ln 7}\) doesn't need any further mathematical simplification. But if you want to get a numerical approximation, you can calculate the values of \(\ln 12\) and \(\ln 7\) and perform the division.

Key Concepts

Natural Logarithms

A natural logarithm is a logarithm with the base of Euler's number, represented as \(e\), which is approximately equal to 2.71828. It is denoted as \(\ln x\). Natural logarithms provide a way to express exponential growth and decay in mathematics and are widely used in calculus and other areas.
The natural logarithm of a number \(x\) is the power to which \(e\) must be raised to equal \(x\). For example, if \(\ln x = 2\), it means \(e^2 = x\). The natural logarithm function is the inverse of the exponential function \(e^x\).
Some important properties of natural logarithms include:

• \(\ln(1) = 0\), because \(e^0 = 1\).
• \(\ln(e) = 1\), because \(e^1 = e\).
• \(\ln(ab) = \ln a + \ln b\), which means the logarithm of a product is the sum of the logarithms.
• \(\ln\left( \frac{a}{b} \right) = \ln a - \ln b\), which means the logarithm of a quotient is the difference of the logarithms.

Change of Base Formula

The change of base formula is a useful tool in mathematics that allows you to rewrite logarithms in terms of natural logarithms or common logarithms (base 10). This formula is especially helpful when a calculator only provides natural or common logarithms, as it allows calculations to be performed using these accessible means.
The change of base formula states: \[\log_b a = \frac{\ln a}{\ln b}\] This equation tells you that any logarithm can be computed by dividing the natural logarithm of the argument \(a\) by the natural logarithm of the base \(b\).

• The formula helps convert less common bases to either base \(e\) (natural logarithm) or base 10 for ease of computation.
• It maintains the same logical relationship since logarithms of different bases are just scaled by a constant factor.
• It is particularly useful in algebra and calculus when simplifying expressions involving logarithms.

For example, given \(\log_7 12\), you can write it as \(\frac{\ln 12}{\ln 7}\) using this formula.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They are often expressed as \(\log_b x\), where \(b\) is the base and \(x\) is the argument. These functions are crucial in solving equations involving exponential growth or decay.
A logarithmic function translates multiplication into addition. That means multiplying numbers is akin to adding their logarithms, which is handy when dealing with large sets of data or calculations in scientific fields.
Key features of logarithmic functions include:

• They have a vertical asymptote on the y-axis, meaning as \(x\) approaches zero, \(\log_b x\) decreases without bound.
• They pass through the point \((1,0)\), since \(\log_b(1)=0\). For base \(e\), this is \((1,\ln 1)\).
• The domain of a logarithmic function is \(x > 0\), as you cannot take the logarithm of a negative number or zero.

Logarithmic scales are used in a variety of applications, such as measuring the intensity of earthquakes (Richter scale), sound (decibels), and pH levels in chemistry. Understanding logarithmic functions is vital for navigating fields that deal with exponential relationships, like biology, economics, and statistics.