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Chapter 4: Problem 29

Solve the following equations for \(x .\) \(\ln (\ln 3 x)=0\)

Short Answer

Expert verified
x = \frac{e}{3}

Step by step solution

01

- Understand the problem

We need to solve for the variable \(x\) in the equation \(\text{ln}(\text{ln} 3x) = 0\). This involves logarithmic functions.
02

- Isolate the inner logarithmic function

To solve for \(x\), we first need to isolate the inner logarithmic function. Recall that \(\text{ln} y = 0\) implies \(y = e^0 = 1\). So, \(\text{ln} 3x = 1\).
03

- Exponentiation to solve for the argument of the inner log

After isolating the inner log, we get \(\text{ln} 3x = 1\). Next, exponentiate both sides to eliminate the natural logarithm: \(3x = e^1 = e\).
04

- Solve for \(x\)

Now, solve the equation \(3x = e\) by dividing by 3: \(x = \frac{e}{3}\).

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Key Concepts

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

solving logarithms
Solving logarithmic equations involves finding the variable's value such that the equation holds true. In this case, we are given \(\ln(\ln 3x) = 0\). To solve this, follow these steps:

  • Identify the outer logarithmic function: \(\ln(\text{something}) = 0\).
  • Recall that \(\ln y = 0\) if and only if \(y = 1\).
  • Set the argument of the outer logarithm equal to 1, giving you \(\ln 3x = 1\).

These steps for solving logarithms involve understanding the basic properties and the relationships between functions.
natural logarithm properties
Understanding natural logarithms is essential. The natural logarithm, denoted as \(\ln\), has several properties:

  • The natural logarithm of 1 is 0: \(\ln 1 = 0\).
  • The natural logarithm is the inverse of the exponential function: if \(\ln y = a\), then \(y = e^a\).
  • Using these properties, we can manipulate and solve logarithmic equations with ease.

For instance, solving \(\ln 3x = 1\) utilizes the property that \(\ln y = a\) implies \(y = e^a\). This is why \(3x = e^1 = e\).
exponentiation steps
Exponentiation is key to solving logarithmic equations. After isolating the inner logarithmic function, we have: \(\ln 3x = 1\).

To clear the logarithm, exponentiate both sides:
\[3x = e^1 = e\].
  • Exponentiating effectively 'cancels out' the \(\ln\) function, converting it back into its base form.
  • This gives a simpler equation, which is often easier to solve.
  • Finally, isolate the variable \(x\) by dividing both sides by 3: \(x = \frac{e}{3}\).

These exponentiation steps demonstrate the principle of applying inverse operations to simplify and solve for the unknown variable.

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Most popular questions from this chapter

Use logarithmic differentiation to differentiate the following functions. \(f(x)=\sqrt[x]{3}\)

Solve the given equation for \(x .\) \(\ln \sqrt{x}-2 \ln 3=0\)

In the study of epidemics, we find the equation $$\ln (1-y)-\ln y=C-r t$$ where \(y\) is the fraction of the population that has a specific disease at time \(t\). Solve the equation for \(y\) in terms of \(t\) and the constants \(C\) and \(r\).

Solve the following equations for \(x .\) . \(5 \ln 2 x=8\)

Which is larger, \(\frac{1}{2} \ln 16\) or \(\frac{1}{3} \ln 27\) ?
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