Problem 55 - Precalculus : Building Concepts and Connections

Question

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log _{2}(x+5)=\log _{2}(x)+\log _{2}(x-3)$$

Step-by-Step Solution

Answer

The solution to the given logarithmic equation, eliminating any extraneous solutions, is \(x = 5\).

Step 1: Introducing the basic principles of logarithm

The first step involves understanding that, in any equation, if the base and the logarithm are the same, then the arguments must also be equal. So the equation \(\log _{2}(x+5)=\log _{2}(x)+\log _{2}(x-3)\) can be written as \(\log _{2}(x+5)=\log _{2}((x)(x-3))\)

Step 2: Applying the logarithmic property

By the logarithmic property, \(\log_ba = \log_bc\) implies \(a = c\). So, the equation transforms into \(x+5 = x^2 - 3x\)

Step 3: Refactoring it into a quadratic equation

The next step is to rewrite the equation as a quadratic equation set to zero. So, it becomes \(x^2 - 4x - 5 = 0\)

Step 4: Factorising the quadratic equation

The factorised form of the equation becomes \((x - 5)(x + 1) = 0\)

Step 5: Solving the factorised equations for x

This can now be solved to give \(x = 5\) or \(x = -1\) as potential solutions

Step 6: Checking for extraneous solutions

The solutions need to be checked for extraneous solutions which make the arguments of the original equation negative. As \(\log_2\) is undefined for negative numbers, \(x = -1\) is an extraneous solution. Therefore, \(x = 5\) is the only valid solution.

Key Concepts

Logarithm Properties

Understanding logarithm properties is fundamental when solving logarithmic equations. A logarithm, defined as \( \log_b(x) \) where \( b \) is the base and \( x \) the argument, is the power to which the base must be raised to obtain the argument. For example, if \( 2^3 = 8 \) then we can say \( \log_2(8) = 3 \).

Some key properties of logarithms that are particularly useful in solving equations include:

• The Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)—This property allows us to convert the logarithm of a product into a sum of logarithms.
• The Quotient Rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)—This is used to transform the logarithm of a division into a difference of logarithms.
• The Power Rule: \( \log_b(m^n) = n \cdot \log_b(m) \)—With this, we can take the exponent outside of the logarithm.

These properties enable us to reshape logarithmic equations in a way that makes them more approachable, particularly by transforming logarithmic expressions into algebraic ones. Leveraging these properties effectively is absolutely essential for solving logarithmic equations correctly.

Solving Quadratic Equations

Quadratic equations are a central topic in algebra, with a standard form of \( ax^2 + bx + c = 0 \), where \( a \) is not zero. Solving quadratics can be approached using several methods, including factoring, completing the square, using the quadratic formula, or graphically using a parabola.

If a quadratic equation can be factored, it becomes very straightforward to solve. The equation is rewritten as \( (x - p)(x - q) = 0 \) where \( p \) and \( q \) are the roots of the equation. According to the zero product property, if the product of two factors is zero, at least one of the factors must be zero. This gives us the solutions \( x = p \) or \( x = q \).

However, not all quadratic equations can be easily factored. In such cases, the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \) can be a reliable and universal solution method, based on the coefficients of the equation. It's important for students to be familiar with all these methods to handle different kinds of quadratic equations they may encounter.

Extraneous Solutions

Extraneous solutions are results that emerge during the process of solving an equation but are not valid solutions to the original equation. These may arise when both sides of an equation are raised to an even power or when applying logarithmic properties.

To identify extraneous solutions, one must always check potential solutions back in the original equation. In the context of logarithmic equations, since logarithms are undefined for negative numbers or zero, solutions that result in a negative argument within the logarithm are considered extraneous.

For example, if solving an equation leads to a negative value for an expression inside a logarithm, this value cannot be correct because it would render the logarithm undefined. Hence, it must be discarded. It's crucial to be vigilant about checking solutions to ensure they are valid, particularly when dealing with logarithms, to avoid erroneously accepting extraneous solutions.