Problem 519 - Precalculus

Question

Condense the expression \(5 \ln (b)+\ln (c)+\frac{\ln (4-a)}{2}\) to a single logarithm.

Step-by-Step Solution

Answer

\( \ln(b^5 c (4-a)^{1/2}) \)

Step 1: Factor out Constants

Recognize that coefficients of logarithms can be expressed as exponents inside the logarithm. This means that the expression can be rewritten as: \(\ln(b^5) + \ln(c) + \ln((4-a)^{1/2})\).

Step 2: Apply Logarithm Addition Property

Combine the logarithms into a single logarithm using the property \( \ln(x) + \ln(y) = \ln(xy) \). This results in: \( \ln(b^5 \cdot c \cdot (4-a)^{1/2}) \).

Key Concepts

Logarithmic Properties

Logarithmic properties are essential tools for simplifying expressions and solving equations involving logarithms. One important property is the ability to convert the product and power relationships with logarithms.

For instance, the **Product Property** states that the logarithm of a product is the sum of the logarithms:

• \( \ln(xy) = \ln(x) + \ln(y) \)

This property allows us to combine multiple logarithms into a single one.

Another valuable property is the **Power Property**, which enables us to factor out constants by writing them as exponents:

• \( a \ln(b) = \ln(b^a) \)

This transformation helps to condense terms and is especially useful when applying further mathematical operations.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: a base and an exponent. The exponent denotes how many times the base is multiplied by itself.

For example, \( b^5 \) means multiplying \( b \) by itself five times. This operation is critical in logarithmic manipulation because logarithms and exponents are inverse functions.

When simplifying logarithmic expressions, we often use exponentiation to switch between multiplicative forms and power forms. This understanding is key when handling logarithmic expressions with coefficients since these coefficients can be expressed as exponents inside the logarithm to simplify the expression further.

Logarithmic Expressions

Logarithmic expressions involve logarithms and can often look complex at first sight. To tackle these effectively:

• Identify the parts of the expression that can be a candidate for applying logarithmic properties.
• Look for opportunities to express constants as exponents or vice-versa.
• Recognize when to use the addition property to combine separate logarithmic parts into one.

Understanding the structure and recognizing patterns are important when working with logarithmic expressions. Over time, with practice, these patterns become easier to identify, streamlining the conversion of complex expressions into simpler forms.

Mathematical Manipulation

Mathematical manipulation refers to the array of techniques used to transform expressions and solve equations. With logarithmic functions, manipulation is often centered on applying properties that allow us to combine or simplify terms.

In our example, transforming each logarithmic part by factoring constants as exponents and using logarithmic identities is a form of mathematical manipulation. These transformations reveal a more straightforward version of an initially complicated expression.

Practicing these manipulations helps in developing skills to tackle a wide range of problems, making it easier to condense and solve logarithmic equations efficiently.