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Chapter 3: Problem 7

Express each equation in logarithmic form. $$32^{3 / 5}=8$$

Short Answer

Expert verified
The logarithmic form of the equation \(32^{\frac{3}{5}} = 8\) is: \( \log_{32}(8) = \frac{3}{5} \).

Step by step solution

01

Identify the base, exponent, and result

In the given equation, we have: _base_ (b): 32 _exponent_ (x): \(\frac{3}{5}\) _result_ (y): 8 Now that we have identified these values, we can rewrite the equation in logarithmic form.
02

Rewrite the equation in logarithmic form

Utilizing the formula we mentioned earlier, the logarithmic form of the given equation can be expressed as: \(\log_b(y) = x\) In this specific case, we have: \(\log_{32}(8) = \frac{3}{5}\) So the logarithmic form of \(32^{3 / 5} = 8\) is: $$\log_{32}(8) = \frac{3}{5}$$

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Key Concepts

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

Understanding Exponents
Exponents are a way to express repeated multiplication of the same number. They consist of a base and an exponent. The base is the number that is being multiplied, while the exponent indicates how many times the base is multiplied by itself. For example, in the expression \(32^{3/5}\), the base is 32, and the exponent is \(\frac{3}{5}\). This fractional exponent means we are dealing with both root and power operations.
  • The numerator of the exponent (3) indicates the power, meaning we take the cube of the root operation.

  • The denominator (5) suggests taking the fifth root of the base.

Both operations combine to transform 32 in such a way that results in 8. Understanding how exponents work with fractional values allows us to explore the flexibility and depth of powers in mathematics.
Deciphering Logarithms
Logarithms are the inverse operations of exponentiation. While exponents deal with multiplying a number by itself, logarithms answer the question: "To what power should we raise a certain base to obtain a specific number?" In the context of the provided equation \(32^{3/5} = 8\), we express the same mathematical relationship using logarithms.
  • The base of the exponent becomes the base of the logarithm.

  • The result of the exponentiation becomes the number inside the logarithm.

  • The exponent itself becomes the result of the logarithmic expression.

Hence, converting \(32^{3/5} = 8\) into logarithmic form, we get \(\log_{32}(8) = \frac{3}{5}\). This form allows us to interpret exponential relationships in a different and often more insightful way.
Equation Transformation: From Exponential to Logarithmic Form
Equation transformation involves rewriting a mathematical expression into a different form that may be more useful for solving a problem. In the case of transforming an exponential equation like \(32^{3/5} = 8\) to a logarithmic one, the process involves switching the components as described in the logarithmic concept.
Breaking it down:
  • Identify the base (b), which, in this case, is 32.

  • Determine the exponent (x), as \(\frac{3}{5}\).

  • Establish the result (y), which is 8.

By recognizing these parts, we can write the equation in logarithmic terms. This highlights the beauty of equation transformation, as it allows for calculations that might otherwise be hidden or complex, making them more straightforward and accessible.

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Most popular questions from this chapter

The number of citizens aged \(45-64 \mathrm{yr}\) is projected to be $$ P(t)=\frac{197.9}{1+3.274 e^{-0.0361 t}} \quad(0 \leq t \leq 20) $$ where \(P(t)\) is measured in millions and \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1990\. People belonging to this age group are the targets of insurance companies that want to sell them annuities. What is the projected population of citizens aged \(45-64 \mathrm{yr}\) in \(2010 ?\)

The alternative minimum tax was created in 1969 to prevent the very wealthy from using creative deductions and shelters to avoid having to pay anything to the Internal Revenue Service. But it has increasingly hit the middle class. The number of taxpayers subjected to an alternative minimum tax is projected to be $$ N(t)=\frac{35.5}{1+6.89 e^{-0.8674 t}} \quad(0 \leq t \leq 6) $$ where \(N(t)\) is measured in millions and \(t\) is measured in years, with \(t=0\) corresponding to 2004 . What is the projected number of taxpayers subjected to an alternative minimum tax in 2010 ?

A function \(f\) has the form \(f(x)=a+b \ln x\). Find \(f\) if it is known that \(f(1)=2\) and \(f(2)=4\).

Sketch the graphs of the given functions on the same axes. \(y=0.5 e^{-x}, y=e^{-x}\), and \(y=2 e^{-x}\)

According to data obtained from the CBO, the total federal debt (in trillions of dollars) from 2001 through 2006 is given by $$ f(t)=5.37 e^{0.07 s t} \quad(1 \leq t \leq 6) $$ where \(t\) is measured in years, with \(t=1\) corresponding to 2001\. What was the total federal debt in 2001 ? In 2006 ?
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