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Chapter 12: Problem 35

Solve. Where appropriate, include approximations to three decimal places. $$ \ln x=5 $$

Short Answer

Expert verified
\( x\approx 148.413 \)

Step by step solution

01

Understand the Equation

The given equation is \(\text{ln} \ x = 5\). This means we need to find the value of \(x\) that satisfies the equation when the natural logarithm of \(x\) equals 5.
02

Rewrite the Equation Using Exponential Form

Recall that the natural logarithm \( \text{ln} \) of a number is the power to which \(e\) (approximately 2.718) must be raised to obtain that number. Therefore, the equation \(\text{ln} \ x = 5\) can be rewritten in its exponential form as: \[ x = e^5 \]
03

Compute the Value

To find the actual value, we need to calculate \(e^5\). Using a scientific calculator: \[ e^5 \approx 148.413 \]

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Key Concepts

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

natural logarithm
A natural logarithm, denoted as \(\text{ln}\), is a special kind of logarithm that has a base of the mathematical constant \(\text{e}\). The constant \(\text{e}\) is approximately equal to 2.718. The natural logarithm of a number is the power to which \(\text{e}\) must be raised to achieve that number.

For example, in the exercise \(\text{ln} \ x = 5\), you are looking for the number \ x\ such that when \(\text{e}\) is raised to the 5th power, we get \ x\. This concept is crucial in solving logarithmic equations, as it allows us to move fluidly between exponential and logarithmic forms.
exponential form
Rewriting the given logarithmic equation in its exponential form helps us solve it more easily. The equation \(\text{ln} \ x = 5\) can be expressed in exponential form because the natural logarithm is based on \(\text{e}\), making the base clear and straightforward.

This transformation turns our logarithmic equation into: \[ x = e^5 \]
Here, \ x\ represents the power of \ e\ (which is approximately 2.718) that results in 5. This special relationship between logarithms and exponents allows for simpler computations and is vital for understanding exponential growth and decay in mathematics and real-world applications.
scientific calculator
To find the value of \ x\ in our transformed exponential equation \ (x = e^5) \, we utilize a scientific calculator. Scientific calculators are equipped with functions to compute powers of \(\text{e}\).

Here's how you can do this on a scientific calculator:
  • Locate the \(\text{e}^x\) button, which raises \ e \ to the power of any number you input.
  • Input \ 5 \ and press the \(\text{e}^x\) button.

The result will be approximately 148.413, as mentioned before. Practicing these calculations can help you become more comfortable with using the exponential function on scientific calculators and solve logarithmic equations efficiently.

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Most popular questions from this chapter

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a} x^{2}-2 \log _{a} \sqrt{x}$$

Use the properties of logarithms to find each of the following. $$\log _{5}(125 \cdot 625)$$

Use the properties of logarithms to find each of the following. $$\log _{2} 16^{5}$$

Solve. $$ \log _{4}(3 x-2)=2 $$

To prepare for Section \(12.5,\) review graphing functions and finding domains of functions. Graph. \(f(x)=\sqrt{x}-3\)
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