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Chapter 4: Problem 103

Solve the equation. Write the solution set with exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{3}\left(\log _{3} x\right)=0\)

Short Answer

Expert verified
The solution set is \(\text{x} = 3\)

Step by step solution

01

Understand the equation

The equation given is \(\text{log}_{3}(\text{log}_{3} x) = 0\). This means that the logarithm of the logarithm base 3 of x is equal to 0.
02

Solve the inner logarithm

First, equate the inner logarithm to another variable. Let \(\text{log}_{3} x = y\)
03

Solve for y

Substitute y back into the equation: \(\text{log}_{3} y = 0\). Since the logarithm of any number to its own base to the zero power is 1, it implies that \(\text{y} = 3^{0}\). Therefore, \(\text{y} = 1\)
04

Substitute back to find x

Replace y with \(\text{log}_{3} x\). So we have \(\text{log}_{3} x = 1\). Since the base is 3, it implies that \(\text{x} = 3^1\). Therefore, \(\text{x} = 3\)
05

Write the solution set

Hence, the exact solution to the equation is \(\text{x} = 3\). Since 3 is an integer, we do not need approximate solutions to four decimal places.

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Key Concepts

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

logarithms
Logarithms are a mathematical way to represent how many times a number, known as the base, must be multiplied by itself to reach another number. For instance, in \(\text{log}_{3} x\), the base is 3 and \(\text{log}_{3} x\) gives us the number of times 3 must be multiplied by itself to produce x.
This is essential in understanding logarithmic equations like the one in our exercise: \(\text{log}_{3}(\text{log}_{3} x) = 0\). Here, \(\text{log}_{3} x\) is the inner logarithmic term.
By understanding how to handle the inner term first, we simplify the entire equation. This step-by-step approach is integral to solving many logarithmic problems.
base change properties
The base change property helps in converting logarithms from one base to another. It鈥檚 useful when the bases aren't the same or aren't convenient.
The formula to change the base is: \(\text{log}_{b} a = \frac{\text{log}_{k} a}{\text{log}_{k} b}\), where b is the current base and k is the new base.
Although the given exercise \(\text{log}_{3}(\text{log}_{3} x) = 0\) doesn't require a base change, understanding the property enhances our toolkit for more complex logarithmic equations.
It helps us simplify problems when dealing with different bases, especially in compound expressions or nested logarithms.
exponential equations
Exponential equations involve expressions where a variable appears in the exponent. An example is \(\text{3}^x = y\), where 3 is the base, x is the exponent, and y is the result.
In our exercise, solving for x involves converting the logarithmic equation to its exponential form: \(\text{log}_{3} x = 1\) becomes \(\text{x} = 3^1\).
This demonstrates how logarithms and exponentials are inverses of each other. Mastery of converting between these forms is crucial for solving equations involving these mathematical concepts.
It shows the relationship between the operations, enabling a deeper understanding and flexibility in problem-solving.

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Most popular questions from this chapter

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{5}\left(\frac{1}{x}\right)=\frac{1}{\log _{5} x} $$

A table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from $$\begin{array}{ll} y=m x+b \text { (linear) } & y=a b^{x} \text { (exponential) } \\ y=a+b \ln x \text { (logarithmic) } & y=\frac{c}{1+a e^{-b x}} \text { (logistic) } \end{array}$$ b. Use a graphing utility to find a function that fits the data. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 3 & 2.7 \\ \hline 7 & 12.2 \\ \hline 13 & 25.7 \\ \hline 15 & 30 \\ \hline 17 & 34 \\ \hline 21 & 44.4 \\ \hline \end{array} $$

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{8} 5 $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(2^{1-6 x}=7^{3 x+4}\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(6 \log _{5}(4 p-3)=18\)
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