Problem 72 - 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

(a) False, (b) False, (c) True, (d) True, (e) False, (f) False, (g) False, (h) True, (i) False, (j) True.

Step 1: Evaluate (a)

The equation \( \log \left(\frac{x}{y}\right)=\frac{\log x}{\log y} \) is claimed. Using the properties of logarithms, we know \( \log \left(\frac{x}{y}\right) = \log x - \log y \). However, \( \frac{\log x}{\log y} \) represents a division and not a subtraction. Thus, this equation is False for all values.

Step 2: Evaluate (b)

The equation \( \log _{2}(x-y)=\log _{2} x-\log _{2} y \) is under consideration. The properties of logarithms state that \( \log _{2}(x-y) eq \log _{2} x-\log _{2} y \). The subtraction of logarithms corresponds to the division of arguments, not the subtraction of arguments. Therefore, this equation is False.

Step 3: Evaluate (c)

Here, \( \log _{5}\left(\frac{a}{b^{2}}\right)=\log _{5} a-2 \log _{5} b \) is evaluated. Using the quotient rule for logarithms, \( \log _{5}\left(\frac{a}{b^{2}}\right)= \log _{5} a - \log _{5} b^2 \). Since \( \log _{5} b^2 = 2\log_{5} b \), the equation is True.

Step 4: Evaluate (d)

The equation \( \log 2^{z}=z \log 2 \) uses the power rule for logarithms, which states \( \log (a^b) = b \log a \). Therefore \( \log 2^z = z \log 2 \) is indeed True.

Step 5: Evaluate (e)

The equation which states \( (\log P)(\log Q) = \log P + \log Q \) is considered. The properties of logarithms do not include multiplication of logs equating to the sum of logs. \( \log P + \log Q \) would rather equal \( \log (PQ) \). Thus, this is False.

Step 6: Evaluate (f)

Consider \( \frac{\log a}{\log b} = \log a - \log b \). The left side is a fraction representing a division of logs, which does not equal the right-side subtraction of logs. This equation simplifies incorrectly for any value of the variables, so it is False.

Step 7: Evaluate (g)

The equation is \( (\log _{2} 7)^{x} = x \log _{2} 7 \). It states that exponentiating a log equals multiplying a constant by a log. This is a misunderstanding of logarithm properties and is False.

Step 8: Evaluate (h)

For \( \log _{a} a^{a}=a \), using the power rule \( \log _{a} a^{b} = b \), hence \( \log _{a} a^{a} = a\log_{a} a = a(1) = a \). Thus, this statement is True.

Step 9: Evaluate (i)

The statement \( \log (x-y)=\frac{\log x}{\log y} \) requires examination. The left represents the log of a difference, while the right is a division of logs rather than any standard log property application. Thus, it is False.

Step 10: Evaluate (j)

The equation \( -\ln \left(\frac{1}{A}\right)=\ln A \) is reformulated as follows: \( -\ln \left(A^{-1}\right) = -(-\ln A) = \ln A \). Using properties of logarithms, this is True.

Key Concepts

Quotient Rule for Logarithms

Understanding the quotient rule for logarithms is essential when dealing with division within logarithmic expressions. This rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Mathematically, it's expressed as:

• \( \log \left( \frac{x}{y} \right) = \log x - \log y \)

This rule reflects the basic property of logarithms related to subtraction. Remember, it's about converting division into subtraction within the log world.
It's crucial to avoid the common mistake of confusing this with the expression \(\frac{\log x}{\log y}\), which is not equivalent.
In the original exercise, understanding this rule allows us to see that equations claiming \( \log \left( \frac{x}{y} \right) = \frac{\log x}{\log y} \) are false, as the latter represents a division of logs and not a subtraction.

Power Rule for Logarithms

The power rule for logarithms simplifies expressions where a number is raised to an exponent inside a logarithm.
This rule states:

• \( \log (a^b) = b \log a \)

This means when you have a power inside a log, you can "bring down" the exponent to the front and multiply it by the logarithm of the base.
For example, in the problem statement \( \log 2^z = z \log 2 \), the power rule is applied, and the equation is true due to this property.
Using this rule helps in simplifying calculations and understanding logarithmic expressions more deeply. It also reduces complex log equations into simpler linear ones.

Logarithm Identities

Logarithm identities are foundational properties used to manipulate and simplify logarithmic expressions.
Some key identities include:

• \(\log_a a = 1\) since the base raised to the power 1 is itself.
• \(\log_a 1 = 0\) because any number raised to the power 0 is 1.

These identities allow you to evaluate logs more easily by reducing them to simpler terms.
In our exercise, an example is \( \log_a a^a = a \), which uses the identity \( \log_a a = 1 \) along with the power rule to confirm the equation is true.
Understanding these identities is crucial as they form the backbone of more complex logarithmic problems. They streamline solving by cutting straight to the values with known results.