Problem 66 - Algebra and Trigonometry

Question

True or False? Discuss each equation and determine whether it is true for all possible values of the variables. (Ignore values of the variables for which any term is undefined.) (a) \(\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}\) (b) \(\log _{2}(x-y)=\log _{2} x-\log _{2} y\) (c) \(\log _{5}\left(\frac{a}{b^{2}}\right)=\log _{5} a-2 \log _{5} b\) (d) \(\log 2^{z}=z \log 2\) (e) \((\log P)(\log Q)=\log P+\log Q\) (f) \(\frac{\log a}{\log b}=\log a-\log b\) (g) \(\left(\log _{2} 7\right)^{x}=x \log _{2} 7\) (h) \(\log _{a} a^{a}=a\) (i) \(\log (x-y)=\frac{\log x}{\log y}\) (j) \(-\ln \left(\frac{1}{A}\right)=\ln A\)

Step-by-Step Solution

Answer

Statements (c), (d), (h), and (j) are True; others are False.

Step 1: Examine Part (a)

Consider the equation \( \log \left(\frac{x}{y}\right)=\frac{\log x}{\log y} \). The left side is a logarithmic identity that can be expressed as \( \log x - \log y \) using the property of logarithms. The right side is a fraction \( \frac{\log x}{\log y} \). These two expressions are not equivalent for all values of \( x \) and \( y \). Hence, this statement is False.

Step 2: Examine Part (b)

For \( \log_{2}(x-y) = \log_{2} x - \log_{2} y \), it would imply \( x-y\) equals \( \frac{x}{y} \), which is generally not true. Therefore, this equation is False for all values of \(x\) and \(y\).

Step 3: Examine Part (c)

Considering \( \log_{5}\left(\frac{a}{b^{2}}\right) = \log_{5} a - 2 \log_{5} b \), we use the properties of logarithms: \( \log_{b}\left(\frac{m}{n}\right) = \log_{b} m - \log_{b} n \) and \( \log_{b}(m^{n}) = n\log_{b} m \). Applying these properties makes the equation true. Thus, it is True.

Step 4: Examine Part (d)

The equation \( \log 2^{z} = z \log 2 \) follows directly from the logarithmic identity \( \log b^{m} = m \log b \). This holds true for all \(z\). Thus, it is True.

Step 5: Examine Part (e)

For \( (\log P)(\log Q) = \log P + \log Q \), we encounter two expressions that involve multiplication on one side and addition on the other with no logarithmic property linking them equivalently. Hence, this is False.

Step 6: Examine Part (f)

\( \frac{\log a}{\log b} = \log a - \log b \) suggests a fractional equality that would only hold should \( \log a = \log b \) or in other contexts irrelevant here. Thus, it is False.

Step 7: Examine Part (g)

The expression \( (\log_{2} 7)^{x} = x \log_{2} 7 \) would imply that raising a logarithm to a power equals multiplying by that power. This isn't a valid property. This statement is False.

Step 8: Examine Part (h)

The statement \( \log_{a}(a^{a}) = a \) is correct. By using \( \log_{b}(b^{m}) = m \), this equation follows the logarithmic power rule. Thus, it is True.

Step 9: Examine Part (i)

For \( \log(x-y) = \frac{\log x}{\log y} \), this implies an equality between a log subtraction and a log division similar to a previous question. This is not a compatible equality property. Hence, it is False.

Step 10: Examine Part (j)

Consider \( -\ln\left(\frac{1}{A}\right) = \ln A \). Using the logarithmic identity that \( \ln\left(\frac{1}{b}\right) = -\ln b \), this equation is a valid transformation. Therefore, this is True.

Key Concepts

Logarithmic Identities

Understanding logarithmic identities is crucial in simplifying and solving equations involving logarithms. These identities relate logs to different mathematical operations:

• Product Rule: The logarithm of a product is equal to the sum of the logarithms. For instance, \( \log_b(x \, y) = \log_b x + \log_b y \).
• Quotient Rule: The logarithm of a division is the difference of the logs: \( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y \).
• Power Rule: The logarithm of a power brings the exponent in front: \( \log_b(x^y) = y \log_b x \).

These identities are valid under all real positive values of the variables involved. However, sometimes equations might be false if they do not correctly apply these properties. For example, in part (a) \( \log\left(\frac{x}{y}\right) = \frac{\log x}{\log y} \), mistakenly uses a division of logs instead of a difference, rendering the equation false.

True or False Statements in Mathematics

Determining the truth of mathematical statements often means dissecting the equation to see if it adheres to mathematical properties and identities. Let's ponder how you decide whether a statement is true or false:

• Check if each side of the equation reflects an established mathematical property.
• Consider special cases or simplified versions of the equation to test validity.
• Use logical reasoning and algebraic manipulations to compare both sides.

For example, take part (e), \( (\log P)(\log Q) = \log P + \log Q \). No logarithmic identity equates a product of logs with their sum, making this false. By attempting to find a context or value where this might work and failing, we can confidently say it does not hold true.

Mathematical Equations Analysis

Analyzing mathematical equations often begins with simplification or transformation using known properties and identities. Here are some techniques:

• Break down complex expressions using fundamental mathematical properties.
• Consider transformation techniques that maintain equality.
• Compare the structure of the equation to known identities or substitution methods.

For analysis, consider part (f): \( \frac{\log a}{\log b} = \log a - \log b \). If we tried assuming possible values, such as \( \log a = \log b \), we see only a specific condition could make this true, and generally, this is false.
Analyzing statements requires clarity on which mathematical rules apply and ensuring both sides transform consistently through valid operations. Consistent practice in analysis can reveal the truth or falsity underlying in mathematical equations.