Chapter 18: Problem 19
Express as a single logarithm with a coefficient of \(1 .\) Assume that the logarithms in each problem have the same base. $$3 \log a-2 \log b+4 \log c$$
Short Answer
Expert verified
The single logarithm with a coefficient of 1 is \( \log(a^3 * c^4 * b^2) \).
Step by step solution
01
Apply the Power Rule
Use the Power Rule of logarithms which states that the expression 'n * log_b(m)' is equivalent to 'log_b(m^n)'. Here, apply this rule to each term: 3 * log(a) becomes log(a^3), -2 * log(b) becomes log(b^(-2)), and 4 * log(c) becomes log(c^4).
02
Apply the Product Rule
Combine the terms with positive coefficients using the Product Rule of logarithms, which combines two logs with addition into a single log representing multiplication of the arguments. Combine log(a^3) and log(c^4) to form log(a^3 * c^4).
03
Apply the Quotient Rule
Use the Quotient Rule of logarithms which states that log_b(m) - log_b(n) = log_b(m/n). Here, apply this rule to log(a^3 * c^4) and log(b^(-2)) to obtain log((a^3 * c^4) / b^(-2)).
04
Simplify the Expression
Simplify the expression by writing b^(-2) as 1/b^2. The single logarithmic expression is log((a^3 * c^4) / (1/b^2)). This further simplifies to log(a^3 * c^4 * b^2) because dividing by a fraction is the same as multiplying by its reciprocal.
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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 key property that allows us to simplify complex logarithmic expressions. It states that a logarithm with an exponent can be rewritten as the exponent times the logarithm of the base number. In mathematical terms, this is expressed as:
\[\begin{equation}{\log_b(m^n) = n \cdot \log_b(m)}\end{equation}\]
For instance, in the given exercise, the term \(3 \log a\) is transformed into \(\log(a^3)\) by applying the power rule. This property is particularly useful for situations where you encounter logarithms multiplied by a scalar. Simplifying logarithms in this way allows you to manage them more efficiently in subsequent calculations and contributes to the overall goal of logarithmic simplification.
\[\begin{equation}{\log_b(m^n) = n \cdot \log_b(m)}\end{equation}\]
For instance, in the given exercise, the term \(3 \log a\) is transformed into \(\log(a^3)\) by applying the power rule. This property is particularly useful for situations where you encounter logarithms multiplied by a scalar. Simplifying logarithms in this way allows you to manage them more efficiently in subsequent calculations and contributes to the overall goal of logarithmic simplification.
Product Rule of Logarithms
When multiplying numbers whose logs are known, the product rule of logarithms comes to the rescue. This rule allows us to take the logarithm of a product of numbers and express it as the sum of their individual logarithms. It is formally written as:
\[\begin{equation}{\log_b(mn) = \log_b(m) + \log_b(n)}\end{equation}\]
In the context of our exercise, after applying the power rule, we have logarithms that are added together: \(\log(a^3)\) and \(\log(c^4)\). Leveraging the product rule, we can combine these logs into a single term, which significantly simplifies the expression to \(\log(a^3 \cdot c^4)\). Through this process, we weave separate logarithmic expressions into a cohesive, more manageable form.
\[\begin{equation}{\log_b(mn) = \log_b(m) + \log_b(n)}\end{equation}\]
In the context of our exercise, after applying the power rule, we have logarithms that are added together: \(\log(a^3)\) and \(\log(c^4)\). Leveraging the product rule, we can combine these logs into a single term, which significantly simplifies the expression to \(\log(a^3 \cdot c^4)\). Through this process, we weave separate logarithmic expressions into a cohesive, more manageable form.
Quotient Rule of Logarithms
Next, let's decode the quotient rule of logarithms, which is just as important as the product rule. It tells us that the logarithm of a quotient can be represented as the subtraction of two logarithms. Mathematically, we state this as:
\[\begin{equation}{\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)}\end{equation}\]
Applying this rule to our exercise, we take the newly formed \(\log(a^3 \cdot c^4)\) and subtract \(\log(b^{-2})\) from it. The resulting expression is \(\log\left(\frac{a^3 \cdot c^4}{b^{-2}}\right)\), bringing us one step closer to a single simplified logarithm. This principle is crucial for breaking down complex logarithms into more digestible pieces and is a key player in the art of logarithmic simplification.
\[\begin{equation}{\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)}\end{equation}\]
Applying this rule to our exercise, we take the newly formed \(\log(a^3 \cdot c^4)\) and subtract \(\log(b^{-2})\) from it. The resulting expression is \(\log\left(\frac{a^3 \cdot c^4}{b^{-2}}\right)\), bringing us one step closer to a single simplified logarithm. This principle is crucial for breaking down complex logarithms into more digestible pieces and is a key player in the art of logarithmic simplification.
Logarithmic Simplification
Logarithmic simplification involves reducing expressions to a less complicated form, often with a lower number of logarithmic terms. It’s about making the expression as straightforward as possible while keeping its value intact. Our exercise demonstrates this by converting a sum and difference of logs into a single logarithm.
After applying the quotient and power rules, the exercise leads us to \(\log((a^3 \cdot c^4) / (1/b^2))\). We can then apply the idea of multiplying by the reciprocal to simplify the fraction, resulting in \(\log(a^3 \cdot c^4 \cdot b^2)\). This process not only simplifies the problem at hand but also equips students with a powerful tool for handling complex logarithmic expressions in various mathematical contexts.
After applying the quotient and power rules, the exercise leads us to \(\log((a^3 \cdot c^4) / (1/b^2))\). We can then apply the idea of multiplying by the reciprocal to simplify the fraction, resulting in \(\log(a^3 \cdot c^4 \cdot b^2)\). This process not only simplifies the problem at hand but also equips students with a powerful tool for handling complex logarithmic expressions in various mathematical contexts.
Logarithmic Expressions
Logarithmic expressions represent the power to which a fixed number (the base) must be raised to produce a given number. They have distinctive properties that make them behave differently from regular algebraic expressions. A sound understanding of these properties enables students to manipulate and simplify logarithms effectively.
In our textbook problem, we've seen these properties in action—power, product, and quotient rules all contribute to reshaping the expression. It's the synergy of these properties that allows us to transform the expression \(3 \log a - 2 \log b + 4 \log c\) into the simplified form \(\log(a^3 \cdot c^4 \cdot b^2)\). Mastery of logarithmic expressions is fundamental in subjects like algebra and calculus, helping students to solve equations that involve exponents and to understand the behavior of exponential and logarithmic functions.
In our textbook problem, we've seen these properties in action—power, product, and quotient rules all contribute to reshaping the expression. It's the synergy of these properties that allows us to transform the expression \(3 \log a - 2 \log b + 4 \log c\) into the simplified form \(\log(a^3 \cdot c^4 \cdot b^2)\). Mastery of logarithmic expressions is fundamental in subjects like algebra and calculus, helping students to solve equations that involve exponents and to understand the behavior of exponential and logarithmic functions.
