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Chapter 4: Problem 4

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

Short Answer

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

Step by step solution

01

Understand the logarithmic form

The given equation is in logarithmic form: \ \( \text{log}_{27} 3 = \frac{1}{3} \). This means that 27 raised to the power of \( \frac{1}{3} \) gives 3.
02

Rewrite as an exponential equation

Convert the logarithmic form \( \text{log}_{27} 3 = \frac{1}{3} \) to its equivalent exponential form: \ \( 27^{\frac{1}{3}} = 3 \).
03

Confirm the exponential equation

Check the correctness by confirming if \( 27^{\frac{1}{3}} = 3 \). We know \( 27 = 3^3 \), so \( 27^{\frac{1}{3}} = (3^3)^{\frac{1}{3}} = 3 \).

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Key Concepts

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

logarithmic form
A logarithmic equation tells you the power to which a base number must be raised to get another number. It's written as \( \text{log}_{b} a = c \), where \( b \) is the base, \( a \) is the result, and \( c \) is the exponent. This means that \( b^c = a \).
In our example, the logarithmic form is \( \text{log}_{27} 3 = \frac{1}{3} \). This means that 27 must be raised to the power \( \frac{1}{3} \) to get 3.
exponential form
The exponential form is simply another way to express the same relationship shown in the logarithmic form. In exponential form, it's written as \( b^c = a \). This shows directly that the base b raised to the power c equals a.
For the given equation, converting \( \text{log}_{27} 3 = \frac{1}{3} \) into exponential form gives us \( 27^{\frac{1}{3}} = 3 \). This expresses the same idea in a different format, showing directly that 27 raised to the power \( \frac{1}{3} \) equals 3.
conversion between logarithmic and exponential equations
Converting between logarithmic and exponential forms helps make different aspects of equations clearer.
To convert from logarithmic to exponential form, remember that \( \text{log}_{b} a = c \) is the same as \( b^c = a \). This means:
  • Rewrite the base \( b \)
  • Make the exponent the value on the other side of the equals sign in the logarithmic form
  • Set it equal to the number next to the base in the logarithmic form

For example, \( \text{log}_{27} 3 = \frac{1}{3} \) converts to \( 27^{\frac{1}{3}} = 3 \).
Similarly, to convert from exponential to logarithmic form, \( b^c = a \) converts to \( \text{log}_{b} a = c \).
The practice of converting between these forms helps solidify your understanding of the relationship between exponents and logarithms.
properties of exponents
Understanding the properties of exponents is crucial for solving equations involving exponents and logarithms.
Here are a few important properties:
  • Product of Powers: \( b^m \times b^n = b^{m+n} \)
  • Quotient of Powers: \( \frac{b^m}{b^n} = b^{m-n} \)
  • Power of a Power: \((b^m)^n = b^{mn} \)
  • Zero Exponent: \( b^0 = 1 \), for any nonzero \( b \)

In our example, we used the property that \((b^m)^n = b^{mn} \). Since \( 27 = 3^3 \), we can write \( 27^{\frac{1}{3}} = (3^3)^{\frac{1}{3}} \). Simplifying, we get \( 3^{3 \times \frac{1}{3}} = 3 \), confirming that \( 27^{\frac{1}{3}} = 3 \).
These properties make it easier to manipulate and solve exponential equations.

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Most popular questions from this chapter

The intensity of a sound is given by $$ I=I_{0} 10^{0.1 L} $$ where \(L\) is the loudness of the sound as measured in decibels and \(I_{0}\) is the minimum intensity detectable by the human ear. a) Find \(I\), in terms of \(I_{0}\), for the loudness of a power mower, which is 100 decibels. b) Find \(I\), in terms of \(I_{0}\), for the loudness of just audible sound, which is 10 decibels. c) Compare your answers to parts (a) and (b). d) Find the rate of change \(d I / d L\). e) Interpret the meaning of \(d I / d L\).

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Differentiate. $$ f(x)=\frac{1}{e^{x}}+e^{1 / x} $$

Differentiate. $$ y=\sqrt{x}+\sqrt{e^{x}}+\sqrt{x e^{x}} $$
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