Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to condense the expression.$$3 \ln x+2 \ln y-4 \ln z$$.

Step-by-Step Solution

Answer

\(\ln \left(\frac{x^3 y^2}{z^4}\right)\)

Step 1: Power Rule

The problem starts with the expression \(3 \ln x + 2 \ln y - 4 \ln z\). Using the power rule of logarithms, which states that \(a \ln b = \ln b^a\), the expression can be rewritten as \(\ln x^3 + \ln y^2 - \ln z^4\).

Step 2: Product Rule

Next, apply the product rule of logarithms, which states that \(\ln a + \ln b = \ln (ab)\), to the first two terms of the expression. The expression becomes \(\ln (x^3 y^2) - \ln z^4\).

Step 3: Quotient Rule

Finally, apply the quotient rule of logarithms, which states that \(\ln a - \ln b = \ln (a/b)\), to the entire expression. This condenses the expression to a single logarithm: \(\ln \left(\frac{x^3 y^2}{z^4}\right)\).

Key Concepts

Power Rule

The power rule for logarithms is a handy tool that simplifies expressions involving logs with coefficients. When you have an expression like \(a \ln b\), you can turn it into \(\ln b^a\). This means that the coefficient in front of the logarithm becomes the exponent inside the log.

• Simply stated, the power rule is: \(a \ln b = \ln b^a\).
• This rule helps you to rewrite logarithmic expressions by moving the coefficient as an exponent.

Think about it like this: if you have \(3 \ln x\), it's equivalent to \(\ln x^3\). This transformation helps you work with the log easily by condensing the expression, preparing it for further simplification.

In our original expression \(3 \ln x + 2 \ln y - 4 \ln z\), we apply the power rule:

• \(3 \ln x \rightarrow \ln x^3\)
• \(2 \ln y \rightarrow \ln y^2\)
• \(-4 \ln z \rightarrow \ln z^4\)

Once each term is transformed, the expression simplifies further when using other logarithm properties.

Product Rule

The product rule of logarithms is a useful technique for combining two logarithmic expressions with the same base into one. This rule states: \(\ln a + \ln b = \ln (ab)\). It illustrates a property that combining two logs with an addition sign is the same as the log of their product.

• Product Rule Formula: \(\ln a + \ln b = \ln (ab)\)
• Combines additions of logs into multiplication of their arguments.

In our exercise after applying the power rule, we end up with \(\ln x^3 + \ln y^2\). Using the product rule:
The expression becomes \(\ln (x^3y^2)\).

This transformation groups the terms together using multiplication, setting the stage for further simplification. The power rule made each individual log simpler, while the product rule combined them into a single logarithmic expression.

Quotient Rule

The quotient rule is another essential property of logarithms, enabling the combination or simplification of logs when one is subtracted from another. The rule states: \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\). When you see subtraction in the logs, you can think of it as division within a single logarithm.

• Quotient Rule Formula: \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\)
• This rule converts subtraction into division within the logarithm.

In our expression, after applying the product rule, we have \(\ln (x^3y^2) - \ln z^4\). Using the quotient rule, this further simplifies to:
\[\ln \left(\frac{x^3y^2}{z^4}\right)\].

Through sequential application of the power, product, and quotient rules, we successfully condensed multiple logarithmic terms into a single expression. Each logarithmic property worked together to simplify and combine the original parts of the expression into a more compact form.