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Chapter 10: Problem 54

Solve each equation. $$\log _{x} 1=0$$

Short Answer

Expert verified
\(x > 0\)

Step by step solution

01

Understand the logarithmic equation

The given equation is \(\text{log}_x 1 = 0\). This is a logarithmic equation where the base is \(x\). Remember that \(\text{log}_b a = c\) implies that \(b^c = a\).
02

Apply logarithm properties

According to logarithmic properties, \(\text{log}_b 1 = 0\) means that any base \(b\) raised to the power of 0 equals 1, that is \(b^0 = 1\).
03

Solve for the base

Since this property tells us that \(x^0 = 1\), it is true for any number \(x\) (except \(x\) cannot be 0 as logarithms are not defined for base 0). Therefore, the solution to the equation is any \(x > 0\).

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Key Concepts

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

logarithm properties
When dealing with logarithmic equations, it's essential to understand the properties of logarithms. One critical property is that \(\text{log}_b 1 = 0\).
This means that any base, let's say \(b\), raised to the power of 0, equals 1. Mathematically, this is expressed as:
\[ b^0 = 1 \]
This property is fundamental to solving many logarithmic equations. Another important property is the change-of-base formula, which allows you to rewrite a logarithm in terms of logs of another base. For example:
\[ \text{log}_b a = \frac{\text{log}_c a}{\text{log}_c b} \]
Properties like these simplify calculations and help in solving logarithmic equations step-by-step.
base of logarithm
The base of a logarithm is the number that is raised to the power of the logarithm. For instance, in \( \text{log}_x 1 = 0 \),
the base is \( x \). Understanding the base is crucial because it dictates the behavior of the logarithmic function.
A logarithm asks, 'To what power must the base be raised, to yield a specific number?'.
For the example given, \[ \text{log}_x 1 \] implies we're looking for the power to which \( x \) must be raised to get 1. This is always 0, since any number raised to the power of 0 is 1.
Some common bases you might encounter include \( 10 \) (common logarithm) and \( e \) (natural logarithm).
solving logarithmic equations
Solving logarithmic equations often involves using logarithm properties. Here are some steps to follow:

  • Identify the logarithmic form: Recognize the given equation and its base.

  • Apply properties: Use logarithm properties to simplify the equation. For example, knowing that \(\text{log}_b 1 = 0\) helps simplify \( \text{log}_x 1 = 0 \).

  • Rewrite in exponential form: Convert the log equation to its exponential counterpart. For \(\text{log}_x 1 = 0\), it translates to \ x^0 = 1 \.

  • Solve for the base: Determine the values of the base that satisfy the equation. Here, any \( x > 0 \) works as \ x^0 = 1 \ for any positive \( x \).


By practicing these steps, you'll become adept at solving logarithmic equations efficiently.

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Most popular questions from this chapter

Revenues of software publishers in the United States for the years \(2004-2016\) can be modeled by the function $$ S(x)=91.412 e^{0.05195 x} $$ where \(x=4\) represents \(2004, x=5\) represents \(2005,\) and so on, and \(S(x)\) is in billions of dollars. Approximate, to the nearest unit, revenue for \(2016 .\) (Data from U.S. Census Bureau.)

How long, to the nearest hundredth of a year, would it take \(\$ 4000\) to double at \(3.25 \%\) compounded continuously?

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{-0.205 x}=9 $$

Why is 1 not allowed as a base for a logarithmic function?

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. See Examples 1-4. $$ \frac{1}{3} \log _{b} x+\frac{2}{3} \log _{b} y-\frac{3}{4} \log _{b} s-\frac{2}{3} \log _{b} t $$
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