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Chapter 9: Problem 4

Write each equation in exponential form. \(\log _{3} 81=4\)

Short Answer

Expert verified
The exponential form is \( 3^4 = 81 \).

Step by step solution

01

Understand the Logarithmic Form

The given equation is \( \log _{3} 81 = 4 \). This represents a logarithmic equation where 3 is the base of the logarithm, 81 is the result, and 4 is the exponent to which the base must be raised to yield the result.
02

Convert to Exponential Form

To convert the logarithmic form to exponential form, recall that \( \log_b{a} = c \) can be rewritten as \( b^c = a \). In this case, \( \log_3{81} = 4 \) becomes \( 3^4 = 81 \).

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Key Concepts

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

Exponential Form
Converting a logarithmic equation into its exponential form is a straightforward yet crucial concept in understanding logarithmic functions.
The exponential form gives us a clear view of what the logarithmic equation is expressing.
In the exponential form, the equation is structured as \( b^c = a \):
  • \( b \) is the base.
  • \( c \) is the exponent.
  • \( a \) is the result.
For the equation \( \log_3{81} = 4 \), converting it to its exponential form, we express it as \( 3^4 = 81 \).
This form tells us that when 3 is raised to the power of 4, it equals 81.
Understanding this switch between forms enriches one's grasp of both the functions and how they relate to each other.
Logarithmic Equation
A logarithmic equation is a useful tool for solving problems involving exponential relationships.
The structure of a logarithmic equation is \( \log_b{a} = c \):
  • \( b \) is the base of the logarithm, which is the number being repeatedly multiplied.
  • \( a \) is the answer we would get from the exponential form.
  • \( c \) is the power or exponent needed for the base to reach the result \( a \).
In our example, \( \log_3{81} = 4 \), we see that the base 3 needs to be raised to the power of 4 to produce 81.
This highlights the inverse relationship logarithms have with exponentials, allowing us to solve for unknowns in various mathematical contexts.
Base of a Logarithm
The base of a logarithm is a fundamental part of understanding how logarithmic functions operate.
The base is the number that is being raised to a power to result in another number and acts as a 'pivot' in handling exponential transformations.
In the equation \( \log_b{a} = c \):
  • \( b \) is the base.
  • These functions tell us how many times we need to multiply the base by itself to achieve a certain number.]
For instance, in \( \log_3{81} = 4 \), the base is 3, indicating how many times 3 must be multiplied together to result in 81.
This concept is vital as it forms the foundation for transitioning between logarithmic and exponential equations effectively.

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Most popular questions from this chapter

Determine whether each function represents exponential growth or decay. $$ y=2(4)^{x} $$

Solve each equation. Round to the nearest ten-thousandth. \(2 e^{x}-5=1\)

An equation for loudness \(L\) in decibels, is \(L=10 \log _{10} R\) where \(R\) is the relative intensity of the sound. Solve \(130=10 \log _{10} R\) to find the relative intensity of a fireworks display with a loudness of 130 decibels.

Solve each equation or inequality. Round to four decimal places. $$ 8.2^{n-3}=42.5 $$

Sketch the graph of each function. Then state the function's domain and range. $$ y=-\left(\frac{1}{5}\right)^{x} $$
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