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Chapter 3: Problem 4

Write each equation in its equivalent exponential form. $$2=\log _{9} x$$

Short Answer

Expert verified
the exponential form of the given logarithmic equation is \(x=81\).

Step by step solution

01

Identify the elements in the logarithmic equation

In the given equation \[2=\log _{9} x\], 9 is the base (b), which is the base of the logarithm, x is the argument of the logarithm and 2 is the value of the logarithm. These correspond to the base, exponent, and result in the exponential form, respectively. The equivalent exponent form based on the logarithmic equation should therefore be: \(b^{y}=x\).
02

Substitute the values into the exponential form

Substitute b=9, y=2 and x into the exponential equation \[b^{y}=x\] which becomes \[9^{2} = x\].
03

Simplify the Exponential Equation

Calculate 9 squared (9^2), which is 81. So the exponential form of the given logarithmic equation is \[x=81\]

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Key Concepts

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

Exponential and Logarithmic Equations
Exponential and logarithmic equations are closely related mathematical expressions. Understanding this connection helps in solving many types of math problems. An exponential equation is typically written in the form
  • \( b^y = x \)
where \( b \) is the base, \( y \) is the exponent, and \( x \) is the result. In contrast, a logarithmic equation expresses the same relationship by asking, "to what exponent must the base \( b \) be raised, to produce \( x \)?" This is written as:
  • \( y = \log_b x \)
To convert between these forms, remember:
  • The logarithm \( \log_b x \) is the exponent \( y \) in the corresponding exponential equation \( b^y = x \).
Such conversions are useful when solving logarithmic equations by turning them into more straightforward exponential equations.
Base of Logarithm
When dealing with logarithms, it's crucial to understand the concept of the base. The base of a logarithm is the number which gets raised to a power. In the logarithmic expression
  • \( \log_b x \)
\( b \) is the base. This base is the foundation of the relationship between exponential and logarithmic equations. For example, if you have \( 2 = \log_9 x \), then 9 is the base. The base determines how the power or exponent transforms to produce the original number \( x \). Some common bases are:
  • 10 (common logarithms)
  • 2 (binary logarithms)
  • \( e \) (natural logarithms, where \( e \approx 2.718 \))
Understanding the base helps you to convert logarithmic equations to exponential forms quickly and solve for unknowns effectively.
Exponential Form
Exponential form is a way to express logarithmic equations in terms of powers. It provides a straightforward method to see how numbers relate. For the logarithmic equation \( 2=\log_9 x \), its exponential form is obtained by identifying the elements:
  • Base \( b \): 9
  • Exponent \( y \): 2
  • Result \( x \): unknown value
The exponential form of the equation becomes:
  • \( 9^2 = x \)
By calculating \( 9^2 \), we find that the value of \( x \) is 81. Writing equations in exponential form often simplifies finding solutions, as it transforms a logarithmic expression into a more direct calculation.

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Most popular questions from this chapter

Use Newton's Law of Cooling, \(T=C+\left(T_{0}-C\right) e^{k t},\) to solve this exercise. At 9: 00 A.M., a coroner arrived at the home of a person who had died. The temperature of the room was \(70^{\circ} \mathrm{F}\), and at the time of death the person had a body temperature of \(98.6^{\circ} \mathrm{F} .\) The coroner took the body's temperature at 9: 30 A.M., at which time it was \(85.6^{\circ} \mathrm{F},\) and again at 10: 00 A.M., when it was \(82.7^{\circ} \mathrm{F} .\) At what time did the person die?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I use the natural base \(e\) when determining how much money I'd have in a bank account that earns compound interest subject to continuous compounding.

What is an exponential function?

In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation.. $$3^{x}=2 x+3$$

a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$ \log _{3} 81, \text { or } \log _{3} 9^{2} ? $$
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