Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, use the one-to-one property of logarithms to solve. \(\ln \left(x^{2}-10\right)+\ln (9)=\ln (10)\)

Step-by-Step Solution

Answer

The solutions are \( x = \frac{10}{3} \) and \( x = -\frac{10}{3} \).

Step 1: Understand the Problem

First, recognize that the equation \( \ln(x^2 - 10) + \ln(9) = \ln(10) \) involves logarithms on both sides. The logarithms are expressions from the natural logarithms, \( \ln \). We will be using the properties of logarithms to solve this problem.

Step 2: Apply the Property of Logarithms

Use the property \( \ln(a) + \ln(b) = \ln(ab) \) to combine the terms on the left-hand side. This gives us \( \ln((x^2 - 10) \cdot 9) = \ln(10) \). Now, the equation is \( \ln(9x^2 - 90) = \ln(10) \).

Step 3: Use the One-to-One Property of Logarithms

According to the one-to-one property of logarithms, if \( \ln(a) = \ln(b) \), then \( a = b \). Hence, set \( 9x^2 - 90 = 10 \).

Step 4: Solve the Resulting Equation

Solve the equation \( 9x^2 - 90 = 10 \). First, add 90 to both sides to get \( 9x^2 = 100 \). Then, divide both sides by 9 to obtain \( x^2 = \frac{100}{9} \).

Step 5: Find the Solution for x

Take the square root of both sides: \( x = \pm \frac{10}{3} \). However, both positive and negative values must be checked to ensure they satisfy the original logarithmic equation.

Step 6: Verify Solutions

Substitute \( x = \frac{10}{3} \) and \( x = -\frac{10}{3} \) back into the original expression \( \ln(x^2 - 10) \): \( (\frac{10}{3})^2 - 10 > 0 \) makes it valid, whereas for \( x = -\frac{10}{3} \) it is also valid because \( (-\frac{10}{3})^2 = (\frac{10}{3})^2 \). Thus, both solutions are valid.

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve logarithms and usually require special methods to solve. A common approach is to use the properties of logarithms, which can help simplify complex equations. When dealing with logarithmic equations, it is important to keep in mind that the base of the logarithms should be the same to utilize properties effectively.

In this exercise, we are specifically working with natural logarithms, which use the base 'e'. These types of equations can often be solved using basic techniques, such as the one-to-one property. This property indicates that if two logarithms with the same base are equal, then their arguments must also be equal. For example, if \( \ln(a) = \ln(b) \), then \( a = b \). Don't forget to check the domain of the logarithmic function, as only positive real numbers can be arguments of logarithms.

Properties of Logarithms

One of the key properties of logarithms is the ability to combine or expand logarithmic expressions to make the equations easier to handle. In this problem, we use the addition property of logarithms, which states \( \ln(a) + \ln(b) = \ln(ab) \). This allows us to combine two logarithmic terms into a single term with a product. By applying this property, the equation \( \ln(x^2 - 10) + \ln(9) = \ln(10) \) simplifies to \( \ln((x^2 - 10) \cdot 9) = \ln(10) \).

This simplification is crucial because it enables the use of the one-to-one property to remove the logarithms and work directly with the arguments \( 9x^2 - 90 = 10 \). It's always important to verify the solutions to ensure they do not lead to invalid logarithmic expressions, such as negative or zero arguments.

Natural Logarithms

Natural logarithms are logarithms with the base \( e \), where \( e \) is approximately equal to 2.718. Denoted as \( \ln \), natural logarithms have unique properties that make them particularly useful in calculus, physics, and many areas of engineering and science. They connect exponential functions with logarithmic functions, providing useful tools for solving equations involving growth and decay.

In our equation \( \ln(x^2 - 10) + \ln(9) = \ln(10) \), we deal solely with natural logarithms. This ensures consistency and allows the use of logarithmic properties like simplification and the one-to-one property. Remember that natural logarithms, like all logarithms, can only accept positive arguments. This constraint means solving logarithmic equations often involves checking that solutions fall within an acceptable range to avoid any undefined expressions.