Problem 97
Question
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
Verified Answer
The given equation is False. The corrected statement should be \(log (x+3)-log (2 x) = log\left(\frac{x+3}{2x}\right)\).
1Step 1: Apply quotient property
We'll rewrite the equation using the quotient property of logarithms. Hence, \(log(x+3) - log(2x)\) becomes \(log\left(\frac{x+3}{2x}\right)\).
2Step 2: Compare the expressions
Now let's compare our result with the right side of the initial equation, which is \(\frac{log(x+3)}{log(2x)}\). Clearly, these expressions are not equal.
3Step 3: Make a correct statement
Let's correct the right side to make it comply with the quotient rule of logarithms. Instead of \(\frac{log(x+3)}{log(2x)}\), it should be \(log\left(\frac{x+3}{2x}\right)\). Now the two sides of the equation are equal.
Key Concepts
Quotient Rule of LogarithmsEquation ValidationMathematical Corrections
Quotient Rule of Logarithms
The quotient rule of logarithms is a fundamental property that helps simplify the expression of the difference between two logarithms.
This rule is particularly useful when dealing with logarithmic equations.
It states that the logarithm of a quotient is the same as the difference between the logarithms of the numerator and the denominator. In simpler terms, when you have \( \log(a) - \log(b) \), you can rewrite it as \( \log\left(\frac{a}{b}\right) \).
This property is extremely powerful for simplifying complex log expressions, making it easier to handle in equations. It is important to remember that both values \(a\) and \(b\) must be positive for the logarithms to be defined. By converting differences of logs into a single log equation, we can more easily manipulate and compare expressions.
For our original exercise, using the quotient rule simplifies \( \log(x+3) - \log(2x) \) into \( \log\left(\frac{x+3}{2x}\right) \). This simplification is essential because it allows us to easily spot discrepancies between expressions—a necessary step when validating equations.
This rule is particularly useful when dealing with logarithmic equations.
It states that the logarithm of a quotient is the same as the difference between the logarithms of the numerator and the denominator. In simpler terms, when you have \( \log(a) - \log(b) \), you can rewrite it as \( \log\left(\frac{a}{b}\right) \).
This property is extremely powerful for simplifying complex log expressions, making it easier to handle in equations. It is important to remember that both values \(a\) and \(b\) must be positive for the logarithms to be defined. By converting differences of logs into a single log equation, we can more easily manipulate and compare expressions.
For our original exercise, using the quotient rule simplifies \( \log(x+3) - \log(2x) \) into \( \log\left(\frac{x+3}{2x}\right) \). This simplification is essential because it allows us to easily spot discrepancies between expressions—a necessary step when validating equations.
Equation Validation
Equation validation is a critical step in checking whether a mathematical equation holds true.
It involves comparing both sides of an equation to determine if they are indeed equal under given conditions.
It's important to carefully analyze the expressions and simplify them where necessary to fully understand their equivalence.
In checking the validity of the equation \( \log(x+3) - \log(2x) = \frac{\log(x+3)}{\log(2x)} \), we applied the quotient rule to the left side.
This simplification helped us visualize the two sides more clearly.
Upon doing this, it's evident that \( \log\left(\frac{x+3}{2x}\right) \) is not the same as \( \frac{\log(x+3)}{\log(2x)} \).
Here, the two sides of the equation are fundamentally different because the right side does not correctly apply logarithmic rules.
Understanding how to validate equations can prevent errors in problem-solving and helps cultivate a clear mindset when handling more complex mathematical expressions. In scenarios like this, identifying discrepancies can guide us in making corrections, which brings us to the next crucial part: making mathematical corrections.
It involves comparing both sides of an equation to determine if they are indeed equal under given conditions.
It's important to carefully analyze the expressions and simplify them where necessary to fully understand their equivalence.
In checking the validity of the equation \( \log(x+3) - \log(2x) = \frac{\log(x+3)}{\log(2x)} \), we applied the quotient rule to the left side.
This simplification helped us visualize the two sides more clearly.
Upon doing this, it's evident that \( \log\left(\frac{x+3}{2x}\right) \) is not the same as \( \frac{\log(x+3)}{\log(2x)} \).
Here, the two sides of the equation are fundamentally different because the right side does not correctly apply logarithmic rules.
Understanding how to validate equations can prevent errors in problem-solving and helps cultivate a clear mindset when handling more complex mathematical expressions. In scenarios like this, identifying discrepancies can guide us in making corrections, which brings us to the next crucial part: making mathematical corrections.
Mathematical Corrections
When an equation is found to be false upon validation, making mathematical corrections is the next logical step.
This involves identifying what went wrong and applying the correct mathematical principles to fix the equation.
In the case of our exercise, the incorrect part was the use of independent quotients of logarithms.
To correct the statement, we revisited the right-hand side of the equation.
Initially, it was given as \( \frac{\log(x+3)}{\log(2x)} \).
To make it true, we adjusted this side by applying the quotient rule of logarithms correctly.
We transformed it to \( \log\left(\frac{x+3}{2x}\right) \). Once both sides of the equation read \( \log\left(\frac{x+3}{2x}\right) \), the equation becomes valid.
Corrections like these are common in mathematics and are essential to ensure equations are sound and logical. It not only involves fixing errors but also learning from them to avoid such mistakes in future problems.
This involves identifying what went wrong and applying the correct mathematical principles to fix the equation.
In the case of our exercise, the incorrect part was the use of independent quotients of logarithms.
To correct the statement, we revisited the right-hand side of the equation.
Initially, it was given as \( \frac{\log(x+3)}{\log(2x)} \).
To make it true, we adjusted this side by applying the quotient rule of logarithms correctly.
We transformed it to \( \log\left(\frac{x+3}{2x}\right) \). Once both sides of the equation read \( \log\left(\frac{x+3}{2x}\right) \), the equation becomes valid.
Corrections like these are common in mathematics and are essential to ensure equations are sound and logical. It not only involves fixing errors but also learning from them to avoid such mistakes in future problems.
Other exercises in this chapter
Problem 96
Solve each equation. $$3|\log x|-6=0$$
View solution Problem 97
Evaluate or simplify each expression without using a calculator. $$e^{\ln 5 x^{2}}$$
View solution Problem 98
Evaluate or simplify each expression without using a calculator. $$e^{\ln 7 x^{2}}$$
View solution Problem 99
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)
View solution