Problem 81 - A Graphical Approach to College Algebra

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 of logarithms states that \( k \log_b(x) = \log_b(x^k) \). We will apply this rule to the expression \( 2 \log_a(z-1) \). This results in \( \log_a((z-1)^2) \).

Step 2: Apply the Product Rule

The product rule of logarithms states that \( \log_b(x) + \log_b(y) = \log_b(xy) \). We will use this rule to combine the logs \( \log_a((z-1)^2) \) and \( \log_a(3z+2) \). This results in a single logarithm: \( \log_a((z-1)^2(3z+2)) \).

Step 3: Final Expression

The expression \( 2 \log_a(z-1) + \log_a(3z+2) \) is now rewritten as a single logarithm: \( \log_a((z-1)^2(3z+2)) \). Here, the logarithm has a coefficient of 1 as required.

Key Concepts

Power Rule of Logarithms

The power rule is a handy shortcut when dealing with logarithms. It's beneficial because it allows us to move coefficients in front of a logarithm into the argument as an exponent. This simplifies complex algebraic expressions and makes equations easier to solve. The formal definition of the power rule is: if you have a situation like \( k \log_b(x) \), you can rewrite it as \( \log_b(x^k) \).

• This transforms the scenario from multiplication with a coefficient to simply working with an exponent.
• This transformation often makes further operations simpler to perform.

To apply this, take the example of \( 2 \log_a(z-1) \). Using the power rule, this expression simplifies to \( \log_a((z-1)^2) \). This simplification is a crucial step as it turns the problem into a single expression, valuable in solving logarithmic equations.

Product Rule of Logarithms

The product rule is another essential property of logarithms. It allows you to combine two separate logarithmic expressions into one, as long as they have the same base.The rule states: \( \log_b(x) + \log_b(y) = \log_b(xy) \).

• This combination is only possible if both logarithms have the same base.
• The rule simplifies the expression by turning addition into multiplication.

For the given task of combining \( \log_a((z-1)^2) \) and \( \log_a(3z+2) \), the product rule was applied, which resulted in the expression \( \log_a((z-1)^2(3z+2)) \).This approach cuts down the complexity by reducing multiple logarithms into one, facilitating easier calculations and comprehension.

Understanding Algebraic Expressions

Algebraic expressions are the foundation of many mathematical concepts, including logarithms. They involve numbers, variables, and operations, such as addition, subtraction, multiplication, and division.In the context of logarithms:

• Algebraic expressions describe the quantities inside the logarithm.
• They are formed by combining constants and variables using arithmetic operations.

In the problem, the expression \((z-1)^2(3z+2)\) is an algebraic expression derived from combining the simplified forms of two logarithmic components.Understanding these expressions illuminates how variables interact. This knowledge is critical in solving for unknowns and in rewriting complex expressions into simpler logarithmic forms that are easier to evaluate or solve.