/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 87 Solve $$ \log _{3} x=-2 $$... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 87

Solve $$ \log _{3} x=-2 $$

Short Answer

Expert verified
\(x = \frac{1}{9}\)

Step by step solution

01

Understand the logarithmic equation

The given equation is \(\text{log}_{3}x = -2\). This is a logarithmic form.
02

Convert to exponential form

Recall that \(\text{log}_{b}a = c\) can be rewritten in exponential form as \(a = b^c\). Here, \(\text{log}_{3}x = -2\) becomes \(x = 3^{-2}\).
03

Evaluate the exponential expression

Calculate \(3^{-2}\). Using the property of exponents, \(3^{-2} = \frac{1}{3^2}\).
04

Simplify the expression

Now, \(3^2 = 9\). Therefore, \( \frac{1}{3^2} = \frac{1}{9} \).

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Key Concepts

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

Solving logarithmic equations
When solving logarithmic equations, the first step is understanding the form of the equation. A logarithmic equation consists of a logarithm set equal to a number, like \(\text{log}_{3}x = -2\). This means that you are finding the value of \(\text{x}\) such that raising the base of the logarithm (in this case, 3) to some power results in that value. Recognizing this form is crucial.
Next, you convert the logarithm to its exponential form to solve for \(\text{x}\). More on converting to exponential form in the next section.
Exponential form
Converting a logarithmic equation to exponential form helps in solving the equation more easily. The general rule for converting is: if you have \(\text{log}_{b}a = c\), then it can be rewritten as \(\text{a} = \text{b}^{c}\).
In our given problem, \(\text{log}_{3}x = -2\) becomes \(\text{x} = 3^{-2}\).
This step is crucial to make further evaluation possible.
By rewriting the logarithm in this way, you can see the relationship between \(\text{x}\), the base (3), and the exponent (-2).
Properties of exponents
Understanding properties of exponents is essential in evaluating expressions once they are converted from logarithmic to exponential form.
From the previous section, we converted our problem to \(\text{x} = 3^{-2}\).
Using the property of negative exponents, we know \(\text{b}^{-n} = \frac{1}{b^{n}}\).
Thus, \(\text{3}^{-2}\) becomes \(\frac{1}{3^{2}}\).
Next, evaluate \(\text{3}^{2}\) which equals 9.
So, \(\frac{1}{3^{2}} = \frac{1}{9}\).
Memorizing and understanding these exponent properties will help you solve various exponential expressions efficiently.

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