Problem 103 - A Graphical Approach to Precalculus with Limits

Question

The following equations are identities because they are true for all real mumbers. Use properties of logarithms to simplify the expression on the left side of the equation so that it equals the expression on the right side, where \(x\) is any neal number. $$\ln |x+\sqrt{x^{2}+3}|+\ln |x-\sqrt{x^{2}+3}|=\ln 3$$

Step-by-Step Solution

Answer

Using properties of logarithms, the given expression simplifies to \( \ln 3 \), confirming the identity.

Step 1: Identify the Logarithm Property

Recall that the sum of two logarithms with the same base, \( \ln a + \ln b \), can be combined into one logarithm by multiplying their arguments: \( \ln(ab) \). Apply this to combine the two logarithms on the left side of the equation.

Step 2: Combine the Logarithms

Combine \( \ln |x+\sqrt{x^2+3}| \) and \( \ln |x-\sqrt{x^2+3}| \) using the property from Step 1: \[ \ln (|x+\sqrt{x^2+3}| \cdot |x-\sqrt{x^2+3}|) \] This expression can be simplified by multiplying the terms inside the absolute value.

Step 3: Simplify the Inside of the Logarithm

Recognize that \( (x+\sqrt{x^2+3})(x-\sqrt{x^2+3}) \) is a difference of squares, which simplifies to: \[ x^2 - (\sqrt{x^2+3})^2 = x^2 - (x^2 + 3) = -3 \] Thus, \( |x^2 - (x^2+3)| = 3 \) since the absolute value of \(-3\) is \(3\).

Step 4: Simplify the Logarithm Expression

Substitute the simplified expression back into the logarithm: \[ \ln 3 \] This matches the expression on the right side of the equation.

Step 5: Verify the Identity

Since both sides of the equation simplify to \( \ln 3 \), the identity holds true for all real numbers \( x \).

Key Concepts

Logarithm Identities

Logarithm identities are essential tools in simplifying expressions involving logarithms. One of the main properties is the sum of logarithms, which states that the log of a product can be expressed as the sum of logs: \( \ln a + \ln b = \ln(ab) \). This property is very helpful when you need to combine multiple logarithms into a single expression.

Another key property is the power rule: \( \ln (a^b) = b \ln a \). This rule allows us to bring exponents down as coefficients, making expressions easier to work with, especially when solving equations.

Additionally, there's the change of base formula, which lets us switch between different logarithm bases: \( \log_b a = \frac{\ln a}{\ln b} \). This is particularly useful when calculators or certain bases are involved. Understanding these identities can greatly ease the process of solving logarithmic equations.

Difference of Squares

The difference of squares is a special algebraic pattern that involves expressions like \( a^2 - b^2 \). This can be factored into the form \((a+b)(a-b)\). Recognizing this pattern simplifies many algebraic manipulations.

In the given exercise, the expression \((x+\sqrt{x^2+3})(x-\sqrt{x^2+3})\) demonstrates the difference of squares. Here, \(a = x\) and \(b = \sqrt{x^2+3}\). This allows us to simplify the product:

• \(x^2 - b^2 = x^2 - (x^2+3)\)
• Which simplifies further to \(-3\), since \(b^2 = x^2 + 3\)

Recognizing the difference of squares helps in reducing complex expressions to simpler ones by identifying patterns that are easy to factor.

Absolute Value Properties

Absolute value properties are crucial when dealing with expressions that may involve negative terms. The absolute value of a number is its non-negative form and is denoted by two vertical bars, like \(|x|\). Simply put, \(|x|\) equals \(x\) if \(x\) is positive or zero, and \(-x\) if \(x\) is negative.

One important property is that for any product inside an absolute value, \(|a \, b| = |a| \, |b|\). This property makes it easier to handle complex arguments within logarithms or other functions.

In the exercise, when \(-3\) is inside the absolute value, it becomes \(|-3| = 3\). This shows how absolute values can remove negative signs, allowing expressions to work within constraints like logarithms, which only accept positive values.