Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the logarithm. Round your result to three decimal places.\(\log _{2} 6\)

Step-by-Step Solution

Answer

After performing the calculation, we find that the \(\log_2(6)\) is approximately 2.585.

Step 1: Setting up the equation

Rewrite the logarithm \(\log_2 6 = x\) as \(2^x = 6\). The problem then translates to finding the power or exponent to which 2 must be raised to get 6.

Step 2: Conversion to Natural Logarithm

Convert the equation into the natural logarithm, which is the logarithm to the base \(e\). The natural logarithm can be used because it is the inverse of the exponential function with base \(e\), and so it can be evaluated on any standard calculator. To do this, take the natural logarithm of both sides, yielding \(\ln(2^x) = \ln(6)\).

Step 3: Simplification and numerical calculation

Using the property of logarithms, the left side of the equation becomes \(x \cdot \ln(2)\). This gives \(x \cdot \ln(2) = \ln(6)\). Solving for \(x\), we get \(x = \ln(6) / \ln(2)\). Compute these values on a calculator to get the numerical result.

Key Concepts

Logarithmic Equations

Understanding logarithmic equations is essential for solving problems that involve finding the power to which a base must be raised to produce a given number. This type of equation typically takes the form of \( \log_b(x) = y \), where \( b \) is the base, \( x \) is the number, and \( y \) is the exponent or power. The equation \( \log_2(6) = x \) is a prime example, where we are seeking the power \( x \) such that \( 2^x = 6 \).

To solve this, we can either use the definition of logarithms or change the base to one that is more convenient, like the natural logarithm. The continuity of the logarithmic function ensures that the values remain proportional when changing from one base to another. So, even when the base is not explicitly mentioned, understanding this concept allows for the use of logarithms in various bases to solve the equation efficiently.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \) is a specific instance of a logarithm where the base is \( e \) – Euler's number, approximately equal to 2.71828. It's a unique base because \( e \) arises naturally in mathematical contexts such as calculus, compound interest, and certain mathematical models involving growth and decay.

In the context of converting \( \log_2(6) \) to a natural logarithm, one uses the identity \( 2^x = e^{\ln(2^x)} \) to transition from base 2 to base \( e \). Using properties of logarithms, \( \ln(2^x) \) simplifies to \( x \cdot \ln(2) \)—highlighting the logarithm of a power rule, which allows pulling the exponent out in front of the logarithm. This method simplifies the computation, as most calculators are equipped to handle natural logarithms directly.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. The generic form is \( b^x \) where \( b \) is the base and \( x \) is the exponent. These functions are the inverse of logarithms and describe situations of rapid growth or decay, like population growth, radioactive decay, and even the intensity of sound.

Logarithmic Properties

Logarithmic properties are rules that make working with logarithms more manageable. One fundamental property is the logarithm of a power, used in simplifying \( \ln(2^x) \) to \( x \cdot \ln(2) \). Another crucial property is the change of base formula, useful when converting logarithms from one base to another, as shown in the solution to our original logarithmic equation.Other properties include the logarithm of a product, which states that \( \log_b(xy) = \log_b(x) + \log_b(y) \), and the logarithm of a quotient, stating \( \log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) \). Understanding these properties allows for the transformation and simplification of complex logarithmic expressions into forms that are easier to evaluate or solve.