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Chapter 4: Problem 81

Write expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers. $$2 \log _{a}(z+1)+\log _{a}(3 z+2)$$

Short Answer

Expert verified
\( \log_a((z+1)^2(3z+2)) \)

Step by step solution

01

Use the power rule of logarithms

Apply the power rule, which states that for any logarithmic expression, \( k \log_b(x) = \log_b(x^k) \). Hence, rewrite \( 2 \log_a(z+1) \) as \( \log_a((z+1)^2) \).
02

Express the sum of logarithms as a single logarithm

Use the product rule, which states that \( \log_b(x) + \log_b(y) = \log_b(xy) \). Add \( \log_a((z+1)^2) \) and \( \log_a(3z+2) \) to get \( \log_a((z+1)^2(3z+2)) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

power rule of logarithms
The power rule of logarithms is a very useful property in logarithms. It states that any logarithmic expression of the form \( k \, \text{log}_b(x) \) can be rewritten as \( \text{log}_b(x^k) \). This essentially means you can move a coefficient in front of the logarithm inside the argument of the logarithm as an exponent.For example:
  • \( 2 \, \text{log}_a(z + 1) \) becomes \( \text{log}_a((z + 1)^2) \)
This makes the expression more compact and is often the first step in simplifying complex logarithmic expressions.
In our exercise, we applied the power rule to convert \( 2 \, \text{log}_a(z + 1) \) to \( \text{log}_a((z + 1)^2) \).
product rule of logarithms
The product rule of logarithms is another beneficial tool. This rule states that the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments. Mathematically, this is expressed as \( \text{log}_b(x) + \text{log}_b(y) = \text{log}_b(xy) \).For example:
  • \( \text{log}_a((z + 1)^2) + \text{log}_a(3z + 2) \) can be combined using the product rule:
  • \( \text{log}_a((z + 1)^2 (3z + 2)) \)
This transformation simplifies the sum of several logarithms into a single logarithmic expression.
In the given exercise, we combined \( \text{log}_a((z + 1)^2) \) and \( \text{log}_a(3z + 2) \) using the product rule.
single logarithm
Writing an expression as a single logarithm can greatly simplify the problem. To achieve this, use rules like the power rule and the product rule, which allow us to combine multiple logarithms into one. This simplifies computations and makes it easier to determine values.
In our exercise, we started with the expression \( 2 \, \text{log}_a(z + 1) + \text{log}_a(3z + 2) \). By applying the power rule first, we converted \( 2 \, \text{log}_a(z + 1) \) to \( \text{log}_a((z + 1)^2) \). Then, using the product rule, we combined this with \( \text{log}_a(3z + 2) \) into a single logarithm \( \text{log}_a((z + 1)^2 (3z + 2)) \).Once an expression is consolidated into a single logarithm, it becomes more manageable and easier to work with in further calculations.

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Most popular questions from this chapter

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