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Chapter 4: Problem 64

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 7+3 \ln x=6 $$

Short Answer

Expert verified
The solution to the given logarithmic equation is approximately \(x \approx 0.717\)

Step by step solution

01

Rearrange the equation

We start by trying isolating the logarithmic term. To do this, we take 7 from both sides of the equation to get: \[ 3 \ln x = 6 - 7 = -1 \]
02

Convert to exponential form

Next, still isolating \(x\), we divide both sides by 3, resulting in: \[ \ln x = -\frac{1}{3} \]. Now we can convert the logarithmic equation into an exponential form. The equation gets converted to: \[ x = e^{(-\frac{1}{3})} \]
03

Compute value using calculator

Finally, we calculate the value of \(x\), letting \(e\) to be approximately 2.71828, to find the decimal representation. Using a calculator for this calculation, we get: \[ x \approx 0.717 \]

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Key Concepts

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

Domain of Logarithmic Functions
When working with logarithmic functions, determining the domain, or valid input values, is essential. For a logarithmic function like \( ln(x)\), the input, \(x\), must always be positive. This requirement stems from the mathematical rule that you cannot take the logarithm of a non-positive number.
Therefore, when solving logarithmic equations, any proposed solution (or value of \(x\)) must be checked to ensure it lies within the domain of the function. In our case, the domain is \(x > 0\).
Failure to ensure the potential solution is within this domain could lead to incorrect or undefined results, which is why this step is crucial in solving these types of problems.
Exponential Form Conversion
To solve equations involving logarithms, converting the logarithmic form to its exponential equivalent is often very useful. This process may simplify the equation greatly.
For example, the equation \( ln(x) = -\frac{1}{3}\) is in logarithmic form. To convert it, remember that if \( ln(b) = a\), it can be rewritten in exponential form as \(b = e^a\).
Applying this to our specific problem, we combine the base \(e\) and the logarithmic expression to find \(x\) as \(x = e^{-\frac{1}{3}}\).
The transformation from logarithmic to exponential form often makes the next steps in solving the equation more straightforward, particularly when calculating the numerical value of the variable.
Decimal Approximation
Once we have an expression in an exponential form, such as \(x = e^{-\frac{1}{3}}\), it may be necessary to find a decimal approximation for everyday use or to provide a final answer in a more accessible format.
Computing a decimal approximation involves using a calculator to ensure precision, especially since dealing with numbers like \(e\) (approximately 2.71828) can lead to lengthy decimals.
When using a calculator, it's important to consider the precision you need. In many cases, solutions are rounded to two decimal places for clarity and succinctness. Therefore, the calculated expression \(e^{-\frac{1}{3}}\) approximates to \(0.72\) when rounded to two decimal places.
These rounded results are often utilized in practical applications where exact numbers are not strictly necessary.

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Most popular questions from this chapter

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use this function to solve. Use an equation to answer this question: How far from the eye of a hurricane is the barometric air pressure 29 inches of mercury? Use the TRACE and ZOOM features or the intersect command of your graphing utility to verify your answer.

Write as a single term that does not contain a logarithm: $$ e^{\ln 8 x^{5}-\ln 2 x^{2}} $$

In parts (a) \(-(c),\) graph \(f\) and \(g\) in the same viewing rectangle. a. \(f(x)=\ln (3 x), g(x)=\ln 3+\ln x\) b. \(f(x)=\log \left(5 x^{2}\right), g(x)=\log 5+\log x^{2}\) \(f(x)=\ln \left(2 x^{3}\right), g(x)=\ln 2+\ln x^{3}\) d. Describe what you observe in parts (a)-(c). Generalize this observation by writing an equivalent expression for \(\log _{b}(M N),\) where \(M>0\) and \(N>0\) e. Complete this statement: The logarithm of a product is equal to ________________________

This group exercise involves exploring the way we grow. - Group members should create a graph for the function that models the percentage of adult height attained by a boy who is \(x\) years old, \(f(x)=29+48.8 \log (x+1) .\) Let \(x=5,6\) \(7, \ldots ., 15,\) find function values, and connect the resulting points with a smooth curve. Then create a graph for the function that models the percentage of adult height attained by a girl who is \(x\) years old, \(g(x)=62+35 \log (x-4)\) Let \(x=5,6,7, \ldots, 15,\) find function values, and connect the resulting points with a smooth curve. Group members should then discuss similarities and differences in the growth patterns for boys and girls based on the graphs.

Explaining the Concepts Describe the shape of a scatter plot that suggests modeling the data with an exponential function.
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