Problem 81 - A Graphical Approach to Precalculus with Limits

Question

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$2 \log _{a}(z-1)+\log _{a}(3 z+2), z>1$$

Step-by-Step Solution

Answer

\( \log_a((z-1)^2(3z+2)) \)

Step 1: Apply the Power Rule

The power rule for logarithms states that \( b \log_a(x) = \log_a(x^b) \). First, apply this rule to the term \( 2 \log_a(z-1) \) to rewrite it as \( \log_a((z-1)^2) \).

Step 2: Combine the Logarithms

Use the product rule for logarithms which states that \( \log_a(x) + \log_a(y) = \log_a(xy) \). Combine \( \log_a((z-1)^2) \) and \( \log_a(3z+2) \) to form \( \log_a((z-1)^2(3z+2)) \).

Step 3: Simplify the Expression

The expression is already simplified as a single logarithm with a coefficient of 1: \( \log_a((z-1)^2(3z+2)) \).

Key Concepts

Power Rule for Logarithms

The power rule for logarithms is a useful property that allows us to simplify the multiplication of a logarithm. This property states that if you have a logarithm of a number raised to an exponent, you can bring the exponent in front as a coefficient. Mathematically, it is expressed as \( b \log_a(x) = \log_a(x^b) \).

This means when you have a coefficient in front of a logarithm, you can "move" it back inside the logarithm as an exponent. For example, in the expression \( 2 \log_a(z-1) \), the power rule lets you rewrite it as \( \log_a((z-1)^2) \). Notice how the '2' becomes the exponent of \((z-1)\). This transformation makes it easier to combine or manipulate the logarithmic expressions further.

It's a great tool for simplifying and manipulating expressions, especially when working towards combining multiple logarithms into a single expression.

Product Rule for Logarithms

The product rule for logarithms helps us combine two logs into one when they involve the same base. This rule is particularly useful when you want to simplify expressions with sums of logarithms by reducing them into a single logarithm.

According to the product rule: \( \log_a(x) + \log_a(y) = \log_a(xy) \). This essentially tells you how to handle the addition of two logarithms of the same base by converting it into the logarithm of the product of the two values.

For example, if you have \( \log_a((z-1)^2) + \log_a(3z+2) \), you would apply the product rule to combine them as \( \log_a((z-1)^2 \cdot (3z+2)) \). We've moved from having two separate logarithms into a single one, which is typically the desired outcome when rewriting expressions.

Rewriting Expressions

Rewriting expressions using logarithm properties is an art of simplification. It involves applying various logarithm rules, such as the power rule and the product rule, to restructure the given expression more efficiently or to a more desirable form.

In exercises involving logarithmic properties, your goal often is to combine multiple terms into a single logarithm. This is achieved by:

• Applying the power rule to handle coefficients in front of logarithms.
• Using the product rule to merge sums of logarithms with the same base.

In the given exercise, after applying these rules, you reframe a complex expression \( 2 \log_a(z-1) + \log_a(3z+2) \) into a simpler form \( \log_a((z-1)^2(3z+2)) \).

Rewriting expressions this way not only simplifies them but also makes evaluating or using them in further calculations more efficient. It’s about seeing the "forest for the trees"—taking individual parts and integrating them into a single understandable expression.