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Chapter 3: Problem 4

Write an equivalent exponential equation. $$\log _{27} 3=\frac{1}{3}$$

Short Answer

Expert verified
The equivalent exponential equation is \( 27^{\frac{1}{3}} = 3 \)

Step by step solution

01

Understanding the Given Logarithmic Equation

The given equation is \(\log_{27} 3 = \frac{1}{3}\). This means that 27 raised to some power is equal to 3.
02

Converting Logarithmic to Exponential Form

To convert the logarithmic equation \(\log_{27} 3 = \frac{1}{3}\) to its exponential form, recognize that the equation states: '27 raised to what power equals 3?' This converts to: \[ 27^{\frac{1}{3}} = 3 \]
03

Simplifying the Exponential Equation

The equation \(27^{\frac{1}{3}} = 3\) can also be understood by expressing 27 as a power of 3. Notice that \(27 = 3^3\), thus substituting this in we get: \[ (3^3)^{\frac{1}{3}} = 3 \]
04

Applying the Power Rule

Apply the power of a power rule \((a^m)^n = a^{mn}\): \[ 3^{3 \times \frac{1}{3}} = 3 \]
05

Simplifying the Exponential Term

Simplify the exponent by multiplying it: \[ 3^{3 \times \frac{1}{3}} = 3^1 \] Finally: \[ 3 = 3 \]

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Key Concepts

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

Logarithmic Equation
A logarithmic equation involves a logarithm with a certain base. In our example, the given equation is \(\log_{27} 3 = \frac{1}{3}\).
This tells us that the power to which 27 must be raised to get 3 is \(\frac{1}{3}\).
Understanding how to work with logarithmic equations is crucial for solving them by conversion.
Exponential Equation
An exponential equation expresses a relationship where a constant base is raised to a variable exponent.
To convert the logarithmic equation \(\log_{27} 3 = \frac{1}{3}\) into an exponential form, we recall the fundamental principle: \(b^y = x\) becomes the exponential equation \(27^{\frac{1}{3}} = 3\).
This means 27 raised to the power of \(\frac{1}{3}\) equals 3.
Power Rule in Exponents
The power rule is a key exponent rule which states that \((a^m)^n = a^{mn}\).
In our example, it's used to simplify \( (3^3)^{\frac{1}{3}}\).
This translates as follows:
  • Step 1: Recognize that 27 can be written as \(3^3 \)
  • Step 2: Apply the power rule \((a^m)^n\ =\ a^{mn}) \) to get \(3^{3 \times \frac{1}{3}} = 3\)
    This simplifies the expression significantly.
Simplifying Exponents
Simplifying exponents involves combining and reducing terms to their simplest form.
Applying the power rule, we obtained \(3^{3 \times \frac{1}{3}} = 3\).
We now simplify the exponent by multiplying: \(3^{3 \times \frac{1}{3}} = 3^1\).
This simplifies to \(3 = 3\), which confirms the equivalent form is correct.
Equivalent Forms
Equivalent forms are different expressions that signify the same value or relationship.
In our problem, we transformed the logarithmic equation \(\log_{27} 3 = \frac{1}{3}\) into its equivalent exponential form \(27^{\frac{1}{3}} = 3\).
Understanding and using equivalent forms helps in seamlessly transitioning between logarithmic and exponential representations, thereby making problem-solving diverse and flexible.

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