Problem 74 - College Algebra and Calculus: An Applied Approach

Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln \sqrt{x-8}=5\)

Step-by-Step Solution

Answer

The solution to the logarithmic equation \(\ln \sqrt{x-8}=5\) is \(x=e^{10}+8\). The exact numerical value, approximated to three decimal places, can be obtained using a scientific calculator or similar tool.

Step 1: Remove the log from the equation

An equation involving a logarithm can be transformed using the property that \(log_b{a^n}=n\). In this form, we no longer have a log but an exponent. The natural logarithm \(\ln\) is a logarithm with base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. Using the above property, we can remove the log and rewrite the equation as: \(e^{5}=\sqrt{x-8}\)

Step 2: Isolate \(x\)

To isolate \(x\), we first square both sides of the equation since we have a square root which we need to eliminate. Squaring both sides of the equation \(e^{5}=\sqrt{x-8}\) gives us \(e^{10}=x-8\). Adding 8 to both sides, we now get: \(x=e^{10}+8\)

Key Concepts

Understanding Natural Logarithm

In mathematics, the natural logarithm is a logarithm with a special base, the mathematical constant e. This constant, approximately equal to 2.71828, is significant because it's the base rate of growth shared by all continually growing processes. The natural logarithm of a number k is written as \(\ln(k)\) and represents the exponent to which e must be raised to obtain the number k.

For instance, if \(\ln(k) = n\), we can apply exponentiation to write it as \(e^n = k\). This relationship shows us how logarithms and exponentiation are inverse operations - the logarithm gives us the exponent as an output, and exponentiation takes the exponent and gives us the original number as an output. It's important to understand this inverse relationship when solving logarithmic equations as it's the foundation for unravelling the variable from the confines of the logarithm.

Properties of Natural Logarithms

Natural logarithms have unique properties that make calculations easier. For instance, \(\ln(1)\) equals 0 because \(e^0 = 1\), and \(\ln(e)\) equals 1 because \(e^1 = e\). When solving equations, recognizing these properties helps to simplify expressions and find solutions more efficiently.

The Role of Exponentiation

Exponentiation is the mathematical operation, written with a base and an exponent, that tells us how many times to multiply the base by itself. For example, \(e^5\) tells us to multiply the constant e by itself 5 times.

In the context of solving logarithms, exponentiation is used to 'reverse' the logarithm. This process transforms a logarithmic equation into an exponential one, thereby making it easier to solve for the unknown variable. As mentioned in the step-by-step solution, the equation involves taking e and raising it to the 5th power to eliminate the natural logarithm and isolate the variable.

Base e in Exponentiation

When the base is the natural number e, the operation is particularly meaningful in natural processes such as continuous compounding in finance or natural growth in biology. The exponential function \(e^x\) is its own derivative, which has profound implications in calculus and the study of change.
Understanding how to work with exponentiation, especially with base e, is crucial in solving not just algebraic equations, but also in comprehending complex real-world scenarios that model exponential growth or decay.

Isolating Variables in Equations

Isolating variables is a fundamental technique in algebra to solve for an unknown number. It involves manipulating the equation to get the variable by itself on one side of the equality. This process may include operations like addition or subtraction, multiplication or division, or even exponentiation and taking roots, depending on the form of the equation.

For the given exercise, our goal is to isolate \(x\). When \(x\) is within a radical, we square both sides to get rid of the square root. After that, it's often a matter of performing basic arithmetic operations to rearrange the terms and completely isolate \(x\) on one side of the equation.

Step by Step Isolation

As seen in the solution, we first squared both sides to eliminate the square root. Following this, the process continued by adding 8 to both sides to isolate \(x\) completely. These steps showcase a systematic approach:

• Square both sides to deal with the radical.
• Perform arithmetic operations to isolate the variable.

Mastering the technique of isolating variables not only aids in solving algebraic problems but also in understanding how to manipulate equations to explore different mathematical concepts.