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Chapter 5: Problem 67

Condense the expression to the logarithm of a single quantity. \(\frac{1}{4} \log _{3} 5 x\)

Short Answer

Expert verified
\The condensed logarithm of a single quantity of the expression \(\frac{1}{4} \log _{3} 5 x\) is \(\log _{3} \sqrt[4]{5x}\)

Step by step solution

01

Identify the Properties of Logarithm

First, it is essential to identify the properties of logarithms that will be useful for this task. The property \(a \cdot \log _{b} c = \log _{b} (c^a)\) is most significant for this exercise. It allows us to move the coefficient of the logarithm as an exponent of the argument.
02

Apply the Logarithm Property

Having identified the logarithm property to use, apply it to the given expression. So, \(\frac{1}{4} \log _{3} 5x\) becomes \(\log _{3} (5x)^{\frac{1}{4}}\).
03

Simplify the Equation

Now, the final step is to simplify the expression in the parenthesis. We can rewrite it without changing its value as \(\log _{3} \sqrt[4]{5x}\). Therefore, \(\log _{3} \sqrt[4]{5x}\) is the logarithm of a single quantity for the given logarithmic expression.

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Key Concepts

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

Properties of Logarithms
Logarithms come with special properties that make them an incredibly handy tool for solving complex equations. One key property is the power rule, which helps you manage expressions more efficiently. The power rule states:
  • When you have a coefficient in front of a logarithm, you can move it as an exponent to the argument inside the log.
  • Mathematically, it is written as: \[ a \cdot \log_b c = \log_b (c^a) \]
Imagine you have \( \frac{1}{4} \log_3 5x \). Using the power rule, you can transform it effortlessly to \( \log_3 (5x)^{\frac{1}{4}} \). Understanding these properties allows you to simplify logarithms and make equations much more manageable. So, whenever you see a coefficient, consider moving it as an exponent inside the logarithm.
Logarithm Transformation
Transforming logarithmic expressions is just about using properties effectively to rewrite expressions. The goal is to condense or expand logarithmic forms to make the problem easier to handle. Typically, transformation involves:
  • Changing coefficients into exponents, using the power rule as seen earlier.
  • Balancing the equation by keeping mathematical integrity intact.
Let's say you start with \( \frac{1}{4} \log_3 5x \). Using logarithmic transformation, that can be rewritten as \( \log_3 (5x)^{\frac{1}{4}} \). This process helps give clarity by reducing terms or combining them into a single entity. Transformation makes your math work clear and concise.
Exponents and Radicals
Exponents and radicals relate closely to how logarithms behave. When dealing with logarithmic expressions, understanding how to handle these two can make a world of difference.
  • An expression like \( (5x)^{\frac{1}{4}} \) signifies an exponent that translates a complex radical expression.
  • The exponent \( \frac{1}{4} \) tells you that we're effectively finding the fourth root of \( 5x \).
  • This means you can also write it as \( \sqrt[4]{5x} \).
Radicals and exponents provide a different view on the same mathematical concept, highlighting how deeply connected these operations are. Recognizing these links allows you to jump between expressions like \( \log_3 (5x)^{\frac{1}{4}} \) and \( \log_3 \sqrt[4]{5x} \), simplifying your work and enhancing problem-solving skills. By mastering exponents and radicals, as they relate to logarithms, you gain a robust toolset for tackling a wide array of mathematical challenges.

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Most popular questions from this chapter

The model $$ t=12.542 \ln \left(\frac{x}{x-1000}\right), \quad x>1000 $$ approximates the length of a home mortgage of $$\$ 150,000$$ at \(8 \%\) in terms of the monthly payment. In the model, \(t\) is the length of the mortgage in years and \(x\) is the monthly payment in dollars (see figure). (a) Use the model to approximate the lengths of a $$\$ 150,000$$ mortgage at \(8 \%\) when the monthly payment is $$\$ 1100.65$$ and when the monthly payment is $$\$ 1254.68$$. (b) Approximate the total amounts paid over the term of the mortgage with a monthly payment of $$\$ 1100.65$$ and with a monthly payment of $$\$ 1254.68$$. (c) Approximate the total interest charges for a monthly payment of $$\$ 1100.65$$ and for a monthly payment of $$\$ 1254.68$$. (d) What is the vertical asymptote for the model? Interpret its meaning in the context of the problem.

\(f(x)=-\log _{3}(-x)\)

In Exercises 23-26, use a calculator to evaluate \(f(x)=\log x\) at the indicated value of \(x\). Round your result to three decimal places. \(x=\frac{4}{5}\)

\(\log _{3} 81=4\)

The table of values was obtained by evaluating a function. Determine which of the statements may be true and which must be false. $$ \begin{array}{|l|l|l|l|} \hline x & 1 & 2 & 8 \\ \hline y & 0 & 1 & 3 \\ \hline \end{array} $$ (a) \(y\) is an exponential function of \(x\). (b) \(y\) is a logarithmic function of \(x\). (c) \(x\) is an exponential function of \(y\). (d) \(y\) is a linear function of \(x\).
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