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Chapter 4: Problem 1

$$ \text { Write an equivalent exponential equation. } $$ $$ \log _{2} 8=3 $$

Short Answer

Expert verified
The equivalent exponential equation is \( 2^{3} = 8 \).

Step by step solution

01

Understand the logarithmic equation

The given logarithmic equation is \ \ \ \( \log_{2} 8 = 3 \). This means that the logarithm of 8 to the base 2 equals 3.
02

Recall the definition of logarithms

The definition of a logarithm states that if \( \log_{b} a = c \), then \( b^{c} = a \). Here, \( b \) is the base, \( a \) is the result, and \( c \) is the exponent.
03

Apply the definition to the given equation

Using the definition from Step 2, rewrite \( \log_{2} 8 = 3 \) as the exponential equation: \[ 2^{3} = 8 \].

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Key Concepts

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

Exponential Equation
An exponential equation involves numbers and variables raised to a power. In simpler terms, it consists of a constant base raised to the power of an exponent. For example, in the equation \(2^3 = 8\), 2 is the base and 3 is the exponent. This equation tells us that if we multiply 2 by itself three times, we get 8.

Here are some key points about exponential equations:
  • An exponential equation has the form \(b^c = a\).
  • The base (b) is the number being multiplied.
  • The exponent (c) tells us how many times to multiply the base by itself.
  • The result (a) is the final number you get after performing the multiplication.
Understanding these basic elements is crucial for solving exponential equations and converting them to different forms like logarithmic equations.
Logarithmic Equation
A logarithmic equation is the inverse of an exponential equation. It helps us find the exponent when the base and the result are known. For example, the logarithmic equation \(\log_{2} 8 = 3\) means: 'To what power must we raise 2 to get 8?' The answer here is 3. This explains why \(\log_{2} 8 = 3\).

Here are some key points about logarithmic equations:
  • A logarithmic equation has the form \(\log_b a = c\).
  • The base (b) is the number being raised to a power.
  • The result (a) is the final number obtained by multiplying the base by itself a number of times.
  • The exponent (c) is the number we are solving for.
Using logarithms can simplify complex problems, especially when dealing with very large or very small numbers.
Base and Exponent
Understanding the roles of base and exponent is critical in both exponential and logarithmic equations. The base is the number that gets multiplied, while the exponent indicates how many times to multiply the base by itself. For example, in \(2^3 = 8\):
  • 2 is the base.
  • 3 is the exponent.
  • 8 is the result.

In logarithmic terms, the base and exponent play the reverse roles. For instance, in \(\log_{2} 8 = 3\), we are asked how many times we have to multiply 2 (the base) to get 8 (the result), and the answer is 3 (the exponent).

Here are some key insights:
  • The base must be a positive number.
  • The exponent can be any real number.
  • When converting from exponential to logarithmic form, remember that the base and the result switch their numerical roles.
This understanding helps you switch between exponential and logarithmic forms easily and solve various equations.

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Most popular questions from this chapter

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