Problem 54 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions. $$2 \ln x-\frac{1}{2} \ln y$$

Step-by-Step Solution

Answer

The condensed logarithmic expression is \(\ln (x^2 y^{\frac{1}{2}})\).

Step 1: Apply the Power Rule

First, transform the the expression according to the power rule. This will give \(2 \ln x = \ln x^2\) and \(-\frac{1}{2} \ln y = \ln y^{-\frac{1}{2}}\). So, the resulting expression is \(\ln x^2 - \ln y^{-\frac{1}{2}}\).

Step 2: Apply the Quotient Rule

Afterwards, apply the quotient rule to combine the two logarithms into one. This will provide \(\ln x^2 - \ln y^{-\frac{1}{2}} = \ln \frac{x^2}{y^{-\frac{1}{2}}}\).

Step 3: Simplify the expression

Finally, simplify the expression in the quotient to its simplest form. Remember that any number to the negative power is equivalent to its reciprocal. Therefore, \(y^{-\frac{1}{2}}\) equals \(\frac{1}{y^{\frac{1}{2}}}\). So, the expression simplifies to \(\ln \frac{x^2}{\frac{1}{y^{\frac{1}{2}}}} = \ln (x^2 y^{\frac{1}{2}})\). This is the single logarithm whose coefficient is 1.

Key Concepts

Logarithmic Expression

A logarithmic expression, often seen as \(\log_b(x)\), represents the power to which the base \(b\) must be raised to produce the number \(x\). When working with such expressions, understanding their properties can greatly simplify handling complex equations.

For example, the expression \(2 \ln x - \frac{1}{2} \ln y\) contains two separate logarithmic terms involving the natural logarithm, where the base \(e\) is implied. The coefficients in front of the logarithms, in this case, 2 and -1/2, suggest that this expression can be manipulated using logarithm properties to condense it into a single term with a coefficient of 1.

Understanding this concept helps in recognizing the underlying structure of complicated logarithmic equations, breaking them down into more manageable pieces, or, as in the exercise provided, combining them into a simpler, more elegant form.

Power Rule for Logarithms

The power rule for logarithms is a property that allows us to manipulate the exponent of the value being logged. It states that \(\log_b(m^n) = n \log_b(m)\), essentially enabling us to move the exponent on the value to the front of the logarithm as a coefficient.

In the exercise \(2 \ln x\), applying the power rule modifies this to \(\ln x^2\). You see, the coefficient 2 is essentially the exponent of \(x\) when moved inside the logarithm. Similarly, for \(\frac{1}{2} \ln y\), applying the power rule gives us \(\ln y^{-\frac{1}{2}}\), reflecting the negative exponent which implies a reciprocal when simplified further.

Example Usage of Power Rule:

• If you have \(3\log_b(x)\), it becomes \(\log_b(x^3)\).
• For \(\frac{1}{3}\ln(z)\), the logarithmic expression simplifies to \(\ln(z^{\frac{1}{3}})\).

By mastering this rule, students can effortlessly reposition exponents and thus simplify logarithmic expressions.

Quotient Rule for Logarithms

The quotient rule for logarithms is a handy property when dealing with the division of two logarithmic expressions with the same base. It states that \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\), meaning that the logarithm of a quotient is the difference of the logarithms.

In our exercise, the quotient rule is used after applying the power rule. We start with the expression \(\ln x^2 - \ln y^{-\frac{1}{2}}\) and apply the quotient rule to combine them into one, \(\ln \frac{x^2}{y^{-\frac{1}{2}}}\). This property is critical for condensing multiple logarithmic terms that would otherwise be tedious to manage separately.

Example Usage of Quotient Rule:

• For \(\log_b(u) - \log_b(v)\), this rule turns it into \(\log_b(\frac{u}{v})\).
• In the case of \(\ln(a) - \ln(b)\), it becomes \(\ln(\frac{a}{b})\).

Grasping the quotient rule allows for streamlining logarithms, making algebraic manipulation and the solving of equations much smoother.