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Chapter 4: Problem 2

$$ \text { Write an equivalent exponential equation. } $$ $$ \log _{3} 81=4 $$

Short Answer

Expert verified
3^4 = 81

Step by step solution

01

- Understand Logarithmic Form

The logarithmic form is given as \( \log_{b}a=c \). In this case, it is provided as \( \log_{3}81=4 \). This means that 3 raised to what power equals 81?
02

- Convert to Exponential Form

Use the definition of a logarithm: \( \log_{b}a=c \) can be converted to the exponential form \( b^{c}=a \). Plug in the values: the base \( b \) is 3, the exponent \( c \) is 4, and the result \( a \) is 81. Thus, \( 3^{4}=81 \).

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Key Concepts

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

Logarithms
Logarithms are an essential concept in mathematics. They help us understand how many times we need to multiply a base number to get another number.

The notation \( \log_{b}a=c \) means that \( b \) raised to the power \( c \) equals \( a \).
For instance, in the equation \( \log_{3}81=4 \), we are saying 3 must be raised to the power of 4 to get 81.

To make it more digestible:
  • The base is 3.
  • The exponent is 4.
  • The result or the number we get is 81.
Remember, logarithms are simply a different way to represent exponents or powers. If you understand exponents, logarithms will make perfect sense!
Exponential Form Conversion
Converting from logarithmic to exponential form is quite straightforward.

The logarithmic form \( \log_{b}a=c \) can be converted to exponential form using the formula: \( b^{c}=a \).
This might look complicated, but it's very simple! Let's break it down:

For the equation \( \log_{3}81=4 \):
  • Base \( b \) is 3.
  • Exponent or power \( c \) is 4.
  • The result \( a \) is 81.
So, \( 3^{4}=81 \). Converting helps you see the relationship in a form you may be more familiar with.

When tackling these conversions, always remember the fundamental relationship between logarithms and exponents.
Mathematics Education
Understanding logarithms and exponential forms are crucial in mathematics education. They are fundamental in various fields such as science, engineering, and finance.

Teaching these concepts effectively involves:
  • Using real-life examples. For example, in finance, logarithms help in calculating compound interest.
  • Breaking down complex ideas into smaller steps.
  • Encouraging students to practice converting between logarithmic and exponential forms.
In essence, these concepts form the foundation for higher-level math and its practical applications.

Always approach learning with patience, and don't be afraid to revisit these core ideas regularly.

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Most popular questions from this chapter

$$ \begin{array}{l} \text { Given } \log _{b} 3=1.099 \text { and } \log _{b} 5=1.609, \text { find each }\\\ \text { of the following. } \end{array} $$ $$ \log _{b} 5 b $$

$$ \text { Write an equivalent logarithmic equation. } $$ $$ 10^{3}=1000 $$

Suppose that \(P_{0}\) is invested in a savings account in which interest is compounded continuously at \(6.5 \%\) per year. That is, the balance P grows at the rate given by $$ \frac{d P}{d t}=0.065 P $$ a) Find the function that satisfies the equation. List it in terms of \(P_{0}\) and \(0.065\). b) Suppose that $$\$ 1000$$ is invested. What is the balance after 1 yr? after 2 yr? c) When will an investment of $$\$ 1000$$ double itself?
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