Problem 92 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. What is wrong with the following step? $$\log x+\log (x+1)=0 \Rightarrow x(x+1)=0$$

Step-by-Step Solution

Answer

The error in the given step is that the logarithmic product rule has not been applied correctly. The right equation after correctly applying the product rule of logarithms should be \(\log [x * (x + 1)] = 0\).

Step 1: Analyze the given steps

We have the given equation \(\log x + \log (x + 1) = 0\) which is falsely simplified as \(x(x + 1) = 0\). The mistake here is that the logarithmic properties are not applied correctly.

Step 2: Correctly applying the properties of logarithms

The correct approach uses the properties of logarithms, specifically the product rule. According to the product rule of logarithms, \(\log x + \log (x + 1)\) should be simplified to \(\log [x * (x + 1)]\). Therefore, the error is in the simplification of the initial equation.

Step 3: Write down the correct equation

The correct equation should therefore be \(\log [x * (x + 1)] = 0\), after correctly applying the properties of logarithms.

Key Concepts

Product Rule of Logarithms

Understanding the product rule of logarithms is crucial when tackling logarithmic expressions involving multiplication. The product rule states that the logarithm of a product of two numbers is equal to the sum of the logarithms of those individual numbers. Mathematically, we express it as \[\begin{equation} \[\log(a \times b) = \log(a) + \log(b)\]\end{equation}\]For instance, if we have \[\begin{equation} \[\log(x) + \log(x+1)\]\end{equation}\]we apply this rule to merge the two logarithms into a single one that logs the product of the inside terms. That gives us: \[\begin{equation} \[\log(x \times (x+1))\]\end{equation}\]This concept is essential for simplification and solving logarithmic equations. Knowing how and when to apply this rule allows us to work with complex logarithmic expressions and prepares us for more advanced mathematical work, such as calculus.

Logarithm Simplification

Logarithm simplification involves applying logarithmic identities and rules to make expressions and equations easier to manage. Apart from the product rule, other important rules include the quotient rule, which deals with division inside the logarithm, and the power rule, used when the argument of the logarithm is raised to a power. Simplifying logarithms correctly is a multi-step process that often requires the use of several rules in conjunction. When simplifying, it's vital to remember that these rules are essentially tools to condense or expand logarithmic expressions at strategic moments during problem-solving. For example, condensing multiple logarithms into one using the product rule makes it easier to isolate the variable and solve for it in equations. Conversely, expanding a complex logarithm into simpler components can be useful in differentiation or integration. It's all about choosing the right tool for the task at hand.

Logarithmic Equations

Solving logarithmic equations is like untangling a knot; we need to carefully unwind the variables that are wrapped up in logarithmic functions. Logarithmic equations contain expressions where the variable is part of a logarithm. To solve these equations, one must typically isolate the logarithmic part of the equation and then exponentiate to remove the log, thereby extracting the variable. However, before reaching for the exponent, the equation must be simplified as much as possible using logarithm rules, as seen in our initial exercise. After simplification, we often get a base logarithm set to zero. Since \[\begin{equation} \[\log_b(a) = 0\]\end{equation}\]implies that \[\begin{equation} \[b^0 = a\]\end{equation}\](where \(b\) is the base of the logarithm), we can usually find the solution by understanding that any base raised to zero equals one. This simplifies our variable search considerably and is the typical next step after applying the required logarithm rules.