Problem 97 - College Algebra

Question

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \log (x+3)-\log (2 x)=\frac{\log (x+3)}{\log (2 x)} $$

Step-by-Step Solution

Answer

The given equation \(\log (x+3)-\log (2 x)=\frac{\log (x+3)}{\log (2 x)}\) is false. The corrected equation to make the statement true is \( \log (x+3) - \log (2x) = \log\left(\frac{x+3}{2x}\right)\).

Step 1: Analyze the given equation

We have been given the equation \(\log (x+3) - \log (2x) = \frac{\log (x+3)}{\log (2x)}\)

Step 2: Apply logarithmic properties

Use the formula \( \log_a(b) - \log_a(c) = \log_a(b/c) \) to simplify the left hand side of the equation. This gives us \( \log\left(\frac{x+3}{2x}\right) = \frac{\log (x+3)}{\log (2x)}\).

Step 3: Evaluate the equation

Now, by comparing both sides we see a contradiction. The LHS is a single logarithm while the RHS is a fraction of two logarithms which are not equivalent. Hence, the given equation is false.

Step 4: Correcting the equation

Based on logarithmic properties, the corrected equation should be \( \log (x+3) - \log (2x) = \log\left(\frac{x+3}{2x}\right)\). Now, both sides will be equal, making the statement true.

Key Concepts

Logarithm Simplification

Understanding how to simplify logarithmic expressions is fundamental for working with logarithms. Simplification often involves reducing complex expressions into their simplest form using logarithmic rules. For example, the subtraction of logarithms with the same base, such as
\(\log(x+3) - \log(2x) \),
can be simplified using one of the logarithm laws: the quotient rule. This law states that the difference of two logarithms with the same base can be written as the logarithm of the quotient of their arguments,
\( \log(\frac{a}{b}) = \log(a) - \log(b) \).
When applying this to an exercise, care must be taken to ensure variables and coefficients are accurately placed within the fraction. A common mistake, like transforming the subtraction of logs incorrectly into the division of logs, leads to confusion and incorrect solutions. Keeping the rules for simplification in mind, always verify if the simplified form of the expression mirrors the intended manipulation of the original log terms. This careful approach to simplification will aid in solving more complex logarithmic equations.

Verifying Logarithmic Equations

When dealing with logarithmic equations, the verification process is key to confirming the correctness of an equation. This involves ensuring that the form of the equation aligns with the underlying properties of logarithms. For instance, in our exercise, we are asked to verify the equation
\( \log(x+3) - \log(2x) = \frac{\log(x+3)}{\log(2x)} \).
This verification step requires us to question whether the right-hand side of the equation accurately reflects the left-hand side, given the applicability of logarithm laws. When the forms don't match up, as is the case with the provided example, we recognize that the equation is false. Thus, verification serves as a safeguard against misapplying logarithmic operations. It ensures each equation represents a true mathematical relationship. To solidify understanding, learners should regularly practice verifying different forms of logarithmic equations, honing the ability to spot inconsistencies and deepen their grasp of log properties.

Logarithm Laws

The laws of logarithms are essential tools for manipulating and understanding logarithmic equations. Some fundamental laws include the product rule, quotient rule, and power rule, all of which are frequently utilized in exercises involving logs. The product rule
\( \log(a) + \log(b) = \log(ab) \)
simplifies the addition of logs into a single log of the product of arguments. Conversely, the quotient rule,
\( \log(\frac{a}{b}) = \log(a) - \log(b) \),
used in our exercise, helps in expressing log subtraction as a single log representing division. Lastly, the power rule
\( \log(a^b) = b\log(a) \)
simplifies logs with exponents. Each law is a building block, enabling the translation of complex logarithmic expressions into simpler, more solvable forms. Mastering these laws not just improves computational efficiency but also deepens conceptual understanding, allowing for greater confidence when approaching logarithmic equations in diverse mathematical contexts.