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Chapter 5: Problem 22

Find all the real-number roots of each equation. In each case, give an exact expression for the root and also (where appropriate) a calculator approximation rounded to three decimal places. $$\ln \left(3 x^{2}\right)=2 \ln (\sqrt{3} x)$$

Short Answer

Expert verified
There are infinitely many solutions: all positive real numbers.

Step by step solution

01

Apply Logarithmic Properties

The given equation is \( \ln(3x^2) = 2 \ln(\sqrt{3}x) \). We first apply the logarithmic identity \( \ln(a^b) = b \ln(a) \) on the right-hand side: \( 2 \ln(\sqrt{3}x) = \ln((\sqrt{3}x)^2) \).
02

Simplify the Right-Hand Side

Simplifying \( \ln\left((\sqrt{3}x)^2\right) \) gives \( \ln(3x^2) \). Thus, the equation simplifies to \( \ln(3x^2) = \ln(3x^2) \).
03

Analyze the Simplified Equation

The equation \( \ln(3x^2) = \ln(3x^2) \) is an identity and holds for any positive value of \( x \), as the logarithm function is defined only for positive arguments.
04

State the Solution

Since the logarithmic equation is true for any \( x > 0 \), the solutions are all positive real numbers. However, since we are looking for expressions and results, \( x \) must satisfy the domain of the original equation, which is \( x eq 0 \).

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Key Concepts

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

Logarithmic Properties
When it comes to solving logarithmic equations, understanding the properties of logarithms is crucial. Logarithms help us by converting multiplicative relationships into additive ones.

Here are some key logarithmic properties that are often used:
  • Product Property: This states that the logarithm of a product is the sum of the logarithms of the factors, i.e., \( \ln(ab) = \ln(a) + \ln(b) \).
  • Quotient Property: The logarithm of a quotient is the difference between the logarithms of the numerator and the denominator: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
  • Power Property: This property is very handy in our current problem, allowing us to take an exponent and bring it out as a multiplier: \( \ln(a^b) = b\ln(a) \).
Using these properties can simplify complex logarithmic expressions and make it easier to solve equations.
Logarithmic Equations
Solving logarithmic equations involves finding the value of the variable that makes the equation true. In our exercise, we started with the equation \( \ln(3x^2) = 2 \ln(\sqrt{3}x) \). The process to simplify it includes:

- Applying Properties: Use the power property to simplify the equation by transforming \( 2\ln(\sqrt{3}x) \) into \( \ln((\sqrt{3}x)^2) \).- Simplification: By expanding \((\sqrt{3}x)^2\) we arrive at \(3x^2\), which simplifies the equation to \( \ln(3x^2) = \ln(3x^2) \).This simplification shows it is an identity, meaning it holds true for any value that fits the logarithm's domain.

Logarithmic equations can often be transformed into a form that allows you to solve for the variable, especially by using inverse properties of logarithms.
Domain of Logarithmic Functions
Understanding the domain of logarithmic functions is essential when solving logarithmic equations. Logarithms of non-positive numbers are undefined in real numbers. For a function \( \ln(x) \), the input \( x \) must always be positive:
  • Positive Argument Requirement: Only positive numbers can be inputs for natural logs, thus, \( x > 0 \).
  • Implication for Equations: When you solve logarithmic functions, ensure that you consider only positive solutions.
  • Identity Solutions: In cases like our problem, where the equation simplifies to an identity, the domain constrains possible values to all positive real numbers except zero, ensuring they remain within the proper domain.
Careful consideration of the domain ensures that the solutions are valid and consistent with the properties of logarithms.

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Most popular questions from this chapter

Solve the inequalities. Where appropriate, give an exact answer as well as a decimal approximation. $$\ln \frac{3 x-2}{4 x+1}>\ln 4$$

Solve each equation. $$3(\ln x)^{2}-\ln \left(x^{2}\right)-8=0$$

Solve the inequality \(\log _{10}\left(x^{2}-6 x-6\right)>0.\)

The half-life of strontium- 90 is 28 years. How much of a \(10-\mathrm{g}\) sample will remain after (a) 1 year? (b) 10 years?

(a) Use a graphing utility to estimate the root(s) of the equation to the nearest one-tenth (as in Example 6). (b) Solve the given equation algebraically by first rewriting it in logarithmic form. Give two forms for each answer: an exact expression and a calculator approximation rounded to three decimal places. Check to see that each result is consistent with the graphical estimate obtained in part (a). $$e^{3 x^{2}}=112$$
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