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Chapter 10: Problem 16

Write in logarithmic form. $$7^{0}=1$$

Short Answer

Expert verified
\(\text{log}_{7}(1) = 0\)

Step by step solution

01

Identify the Components of the Exponential Form

The given equation is in the form of an exponential equation: \(7^{0}=1\). In this form, 7 is the base, 0 is the exponent, and 1 is the result.
02

Recall the Logarithmic Form

The logarithmic form of an equation relates the base, the exponent, and the result. It is written as: \(\text{log}_{\text{base}}(\text{result}) = \text{exponent}\).
03

Write the Logarithmic Equation

Substitute the identified components into the logarithmic form: \(\text{log}_{7}(1) = 0\).

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Key Concepts

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

Understanding Exponential Equations
An exponential equation is a mathematical expression where a constant base is raised to a variable exponent. In the exercise, the given exponential equation is: \(7^{0} = 1\). Here:
  • 7 is the base,
  • 0 is the exponent,
  • and 1 is the result.
The equation states that raising the base (7) to the power of the exponent (0) results in 1. This highlights an interesting property: any non-zero number raised to the power of 0 is always equal to 1. Breaking down exponential equations like this helps in understanding and converting them into logarithmic form.
Converting to Logarithmic Equations
Logarithmic equations allow us to express an exponential equation in another form. Essentially, logarithms are the inverse operations of exponentiation. The general form of a logarithmic equation is: \( \text{log}_{\text{base}}(\text{result}) = \text{exponent} \). To convert our example, \(7^{0} = 1\), to logarithmic form, we identify:
  • The base: 7
  • The result: 1
  • The exponent: 0
Substituting these values into the logarithmic form transforms the equation to: \( \text{log}_{7}(1) = 0 \). This shows that the exponent (0) is what you raise the base (7) to, in order to get the result (1). Understanding this transformation is key in solving and comprehending logarithmic equations.
Base-Exponent Relationship
The relationship between the base and the exponent is fundamental in both exponential and logarithmic forms. In the equation \(7^{0} = 1\), the base (7) and the exponent (0) interact to give the result (1). This relationship can be interpreted through logarithms, explaining that the logarithm of 1 with any base is always 0, which can be denoted as: \( \text{log}_{7}(1) = 0 \). Key points to remember:
  • The base in an exponential form becomes the base in the corresponding logarithmic form.
  • The result of the exponentiation becomes the argument of the logarithm.
  • The exponent in the original equation becomes the result of the logarithmic equation.
Recognizing these relationships makes it easier to switch between exponential and logarithmic forms, revealing how closely these mathematical concepts are intertwined.

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Most popular questions from this chapter

The concentration of a drug in a person's system decreases according to the function $$ C(t)=2 e^{-0.125 t} $$ where \(C(t)\) is in appropriate units, and \(t\) is in hours. Approximate answers to the nearest hundredth. (a) How much of the drug will be in the system after \(1 \mathrm{hr} ?\) (b) How long will it take for the concentration to be half of its original amount?

Graph each logarithmic function. $$f(x)=\log _{4} x$$

How much money must be deposited today to amount to \(\$ 1850\) in \(40 \mathrm{yr}\) at \(6.5 \%\) compounded continuously?

Solve each equation. Give exact solutions. $$ \log _{4}(x-3)^{3}=4 $$

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{-x}=\pi $$
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