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Chapter 12: Problem 6

Write in logarithmic form. \(3^{6}=729\)

Short Answer

Expert verified
\(\text{log}_{3}(729) = 6\)

Step by step solution

01

Identify the Components

First, identify the base, the exponent, and the result in the given exponential equation. In the equation \(3^{6}=729\), the base is 3, the exponent is 6, and the result is 729.
02

Understand the Logarithmic Form

The logarithmic form is written as \(\text{log}_{\text{base}}(\text{result}) = \text{exponent}\). This transforms the given exponential form into a logarithmic equation.
03

Convert to Logarithmic Form

Now, apply the transformation: the base 3, result 729, and exponent 6 from \(3^{6}=729\). The logarithmic form will be \(\text{log}_{3}(729) = 6\).

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Key Concepts

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

Base in Logarithmic Form
Understanding the base is crucial when working with exponential and logarithmic equations. In the context of the exercise given, the base is the number that is raised to the power of the exponent. For the equation provided, \(3^6 = 729\), the base is 3.

The base remains the same in both exponential and logarithmic forms. When converting from exponential to logarithmic form, like in our example, the base 3 in \(3^6 = 729\) stays the same: \(\text{log}_3(729) = 6\).

This continuity of the base helps maintain the relationship between the numbers involved. The base can be any positive number other than 1.
Exponent in Logarithmic Form
The exponent in an exponential equation represents the power to which the base is raised. In the equation \(3^6 = 729\), the exponent is 6.

When converting to logarithmic form, the exponent becomes the value on the other side of the equation. In our transformation, the 6 in \(3^6 = 729\) appears as the result in \(\text{log}_3(729) = 6\).

This shows the critical link between the exponential and logarithmic forms. They relay the same relationship but from different perspectives. The exponent answers the question: 'To what power must the base be raised to get this result?' This is key to the conversion process.
Result in Logarithmic Form
The result in an exponential equation is the outcome of the base raised to the exponent. For \(3^6 = 729\), the result is 729.

In logarithmic form, this result is used inside the logarithm. It answers the question: '3 raised to what power equals 729?' This is shown in our equation \(\text{log}_3(729) = 6\).

Understanding the result helps ground the practical application of logarithms. It's essential in real-world calculations, enabling us to solve problems involving exponential growth or decay, such as population growth, radioactive decay, and interest calculations.

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Most popular questions from this chapter

Each of the following functions is one-to-one. Graph the function as a solid line (or curve), and then graph its inverse on the same set of axes as a dashed line (or curve). Complete any tables to help graph the functions. $$ f(x)=x^{3}-2 $$

The number of paid music subscriptions (in millions) in the United States from 2010 to 2016 can be modeled by the exponential function $$ f(x)=1.365(1.565)^{x} $$ where \(x=0\) represents \(2010, x=1\) represents \(2011,\) and so on. Use this model to approximate the number of paid music subscriptions in each year, to the nearest thousandth. (Data from RIAA.) (a) 2010 (b) 2013 (c) 2016

Use the special properties of logarithms to evaluate each expression. \(\log _{3} 3\)

Use the special properties of logarithms to evaluate each expression. \(\log _{8} 8\)

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{0.45 x}=\sqrt{7}$$
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