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Chapter 4: Problem 29

Write the equation in logarithmic form. $$ \left(\frac{1}{5}\right)^{-3}=125 $$

Short Answer

Expert verified
log_{\frac{1}{5}} (125) = -3.

Step by step solution

01

- Identify the base, exponent, and result

The given equation is \(\frac{1}{5}^{-3}=125.\)Here, the base is \(\frac{1}{5},\)the exponent is -3, and the result is 125.
02

- Understand Logarithmic Form

To convert an equation \( a^b = c \) to logarithmic form, write \( \text{log}_a(c) = b \).
03

- Apply the Logarithmic Form

Using the information identified, convert \( \frac{1}{5}^{-3} = 125 \) to logarithmic form: \( \text{log}_{\frac{1}{5}} (125) = -3 \).

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Key Concepts

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

Exponent
In mathematics, an exponent refers to the number that indicates how many times a base is multiplied by itself. For example, in the expression \(a^b\), the base \(a\) is raised to the power of the exponent \(b\).
Exponents are powerful tools for representing large numbers in a compact form. They often appear in scientific notation or while describing phenomena that grow exponentially.
In our given exercise, the exponent is -3. This tells us that we are multiplying the fraction \( \frac{1}{5} \) by itself three times and taking the reciprocal because the exponent is negative.

Here are some key points to remember about exponents:
  • Positive exponents indicate multiplication of the base.

  • Negative exponents represent division or taking the reciprocal of the base.

  • An exponent of zero means the result is always 1 (except for 0 raised to the power of 0, which is indeterminate).
  • Fractional exponents indicate roots. For example, \(a^{\frac{1}{2}}\) represents the square root of \(a\).
Base
A base in an exponential expression is the number that is multiplied by itself. In the equation \(a^b = c\), \(a\) is the base. The base is a fundamental component and dictates the fundamental value being repeatedly multiplied.
In the given exercise, the base is the fraction \( \frac{1}{5} \). This means that the fraction \( \frac{1}{5} \) is the value that we multiply by itself repeatedly according to the exponent value.

Here are some important points about bases:
  • The base can be any real number, positive or negative.

  • A base of 1, regardless of the exponent, always results in 1.

  • Understanding the base is crucial for solving both exponential and logarithmic equations.
Logarithm
A logarithm is the inverse operation of exponentiation. It asks the question:

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Most popular questions from this chapter

Explain how to use the change-of-base formula and explain why it is important.

Which functions are exponential? a. \(f(x)=\left(\frac{1}{\sqrt{3}}\right)^{x}\) b. \(f(x)=1^{x}\) c. \(f(x)=x^{\sqrt{3}}\) d. \(f(x)=(-2)^{x}\) e. \(f(x)=\pi^{x}\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log \left(x^{2}+7 x\right)=\log 18\)

A table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from $$\begin{array}{ll} y=m x+b \text { (linear) } & y=a b^{x} \text { (exponential) } \\ y=a+b \ln x \text { (logarithmic) } & y=\frac{c}{1+a e^{-b x}} \text { (logistic) } \end{array}$$ b. Use a graphing utility to find a function that fits the data. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 3 & 2.7 \\ \hline 7 & 12.2 \\ \hline 13 & 25.7 \\ \hline 15 & 30 \\ \hline 17 & 34 \\ \hline 21 & 44.4 \\ \hline \end{array} $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(21,000=63,000 e^{-0.2 t}\)
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