Question

Which of the following are valid? \((a) \ln u-2=\ln \frac{u}{e^{2}}\) (b) \(3+\ln v=\ln \frac{e^{3}}{v}\) (c) \(\ln u+\ln v-\ln w=\ln \frac{u v}{w}\) \((d) \ln 3+\ln 5=\ln 8\)

Short answer

(a) and (c) are valid.

Step-by-step solution

Understand the Problem

We need to determine whether each expression follows the rules of logarithms. Each expression involves comparing two logarithmic statements, so we'll check for equivalency using logarithmic identities.

Analyze Expression (a)

Expression (a) is \( \ln u - 2 = \ln \frac{u}{e^{2}} \). Using the property \( \ln a - b = \ln \frac{a}{e^b} \), we find: \( \ln u - 2 = \ln \frac{u}{e^{2}} \), so this expression is valid.

Analyze Expression (b)

For expression (b), \( 3 + \ln v = \ln \frac{e^{3}}{v} \) can be rearranged using the property \( b + \ln a = \ln (e^b a) \). Then, we have \( 3 + \ln v = \ln (e^3 v) \), which does not match the right hand side, hence this expression is invalid.

Analyze Expression (c)

Expression (c), \( \ln u + \ln v - \ln w = \ln \frac{uv}{w} \), follows the property \( \ln a + \ln b - \ln c = \ln \frac{ab}{c} \). Thus, this expression is valid.

Analyze Expression (d)

For expression (d), \( \ln 3 + \ln 5 = \ln 8 \). Using the property \( \ln a + \ln b = \ln (a \cdot b) \), \( \ln 3 + \ln 5 = \ln (3 \cdot 5) eq \ln 8 \). Therefore, this expression is invalid.

Key concepts

Properties of Logarithms

Logarithms have specific properties that help simplify numerous mathematical expressions. One important property is the **product rule**: - The product rule states that the logarithm of a product is equal to the sum of logarithms of individual factors. In terms of logarithms, this rule is represented as \( \ln(a \cdot b) = \ln a + \ln b \). Another essential property is the **quotient rule**: - The quotient rule declares that the logarithm of a ratio is the difference between the logarithms of the numerator and the denominator. This can be expressed as \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \). Furthermore, we have the **power rule**: - The power rule asserts that the logarithm of a number raised to some power is equal to the power multiplied by the logarithm of the number. Mathematically, this is stated as \( \ln(a^b) = b \cdot \ln a \). Understanding these core properties helps you easily manipulate logarithmic expressions and determine whether they are valid or not.

Equivalence of Expressions

Determining if two logarithmic expressions are equivalent involves comparing them using the properties mentioned above. This equivalence implies that both expressions ultimately compute the same value for any given input that lies within their domain. For example, consider expression (a): \( \ln u - 2 = \ln \frac{u}{e^{2}} \). Using the quotient rule, this expression states: - Using the power rule, \( -2 \) can be interpreted as \( -\ln(e^2) \), which simplifies to \( \ln \left( \frac{u}{e^2} \right) \). When evaluating expression (c): \( \ln u + \ln v - \ln w = \ln \frac{uv}{w} \), we utilize both product and quotient rules, confirming that the initial expression is equivalent to the simplified one. Being able to spot and articulate equivalences between logarithmic expressions is a valuable skill in mathematics. It's crucial for resolving complex equations and verifying the mathematical validity of identities.

Mathematical Validity

Mathematical validity checks whether expressions or equations align with established rules. In this context, it involves ensuring logarithmic expressions are true by verifying with logarithmic properties. Take expression (b) \( 3 + \ln v = \ln \frac{e^{3}}{v} \). Here, applying the power rule converts \( 3 \) into \( \ln(e^3) \), suggesting the expression should read \( \ln(e^3 v) \). The right-hand side shows \( \ln \left( \frac{e^3}{v} \right) \), which doesn't equate to the left-side expression, hence it's invalid. Similarly, expression (d), \( \ln 3 + \ln 5 = \ln 8 \), violates the product rule, as \( \ln(3 \cdot 5) = \ln 15 \) not \( \ln 8 \). Ensuring mathematical validity means aligning procedures and results with established mathematical principles. This careful verification process safeguards against errors in mathematical reasoning and outcomes.