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Chapter 12: Problem 46

Solve. Where appropriate, include approximations to three decimal places. $$ 3 \ln x=-3 $$

Short Answer

Expert verified
x \approx 0.367

Step by step solution

01

Isolate the logarithmic expression

Divide both sides of the equation by 3 to isolate the natural logarithm. \[ \frac{3 \ln x}{3} = \frac{-3}{3} \] This simplifies to: \[ \ln x = -1 \]
02

Convert the logarithmic equation to an exponential equation

Use the definition of the natural logarithm to rewrite the equation in its exponential form. \[ e^{\ln x} = e^{-1} \] This simplifies to: \[ x = e^{-1} \]
03

Compute the approximate value

Use a calculator to find the approximate value of \( e^{-1} \).\[ e^{-1} \approx 0.367 \]

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Key Concepts

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

isolating logarithms
To solve equations involving logarithms, the first crucial step is to isolate the logarithmic expression. This helps simplify the problem and makes further steps more manageable. In the problem given, we started with the equation:
3 \ln x = -3.
To isolate \ln x, we need to get rid of the coefficient 3 in front of it.
We do this by dividing both sides of the equation by 3:
\( \frac{3\ln x}{3} = \frac{-3}{3} \).
This action isolates the logarithm, simplifying it to:
\( \ln x = -1 \).
Isolating logarithms sets the stage for converting the equation into an exponential form, a crucial next step in solving the equation.
exponential form
After isolating the logarithm, the next step is to convert the logarithmic equation to its exponential form. This is based on the equivalence between logarithmic and exponential expressions. Specifically, if you have \( \ln x = y \), you can rewrite it using the base of natural logarithms:
\( e^{\ln x} = e^y \).
In our specific problem, we have:
\(\ln x = -1\)
So, we convert it to exponential form:
\( e^{\ln x} = e^{-1} \).
This simplifies to:
\( x = e^{-1} \).
Knowing the relationship between logarithms and exponentials is essential for solving equations like this one. It allows us to undo the logarithm and find the value of the variable.
approximation
The final step in solving the equation is to get an approximate numerical value. The exponential expression \( e^{-1} \) is not often intuitive. So, we use a calculator to find its approximate value.
Plugging \( e^{-1} \) into the calculator, we get:
\( e^{-1} \approx 0.367 \).
When asked to provide approximations up to three decimal places, this kind of precision is typically required. Therefore, it鈥檚 vital to follow proper rounding rules for accuracy. The ability to switch between exact values and their decimal approximations is useful not just for homework problems, but also for real-world applications where exact values might not be practical or necessary.

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Most popular questions from this chapter

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a}\left(x^{2}-4\right)-\log _{a}(x+2)$$

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a}\left(x^{8}-y^{8}\right)-\log _{a}\left(x^{2}+y^{2}\right)$$

Simplify. $$ \frac{a^{15}}{a^{3}} $$

Given \(\log _{b} 3=0.792 \text { and } \log _{b} 5=1.161\). If possible, use the properties of logarithms to calculate numerical values for each of the following. $$\log _{b} 45$$

Explain why we say that "a logarithm is an exponent"
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