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Chapter 4: Problem 21

Write each equation in exponential form. $$\log _{e}(54.598) \approx 4$$

Short Answer

Expert verified
The exponential form is \( e^4 = 54.598 \).

Step by step solution

01

Understand the logarithmic form

The equation given is \( \log_e(54.598) \approx 4 \), which is in the form of \( \log_b(a) = c \). Here, the base \( b \) is \( e \), \( a \) is \( 54.598 \), and \( c \) is approximately \( 4 \).
02

Convert to exponential form

The exponential form of a logarithmic equation \( \log_b(a) = c \) is \( b^c = a \). By substituting the values from Step 1 into this formula, we get \( e^4 = 54.598 \).
03

Confirm the conversion

Verify that changing the equation from logarithmic form to exponential form is consistent. Calculate \( e^4 \) and check if it approximately equals \( 54.598 \). Since \( e^4 = 54.598 \) does hold true, the conversion is correct.

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Key Concepts

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

Logarithmic Equations
Logarithmic equations are essential for connecting exponential relationships with more complex problems. They are written in the form \( \log_b(a) = c \). In such equations:
  • \( b \) is the base of the logarithm. It tells us what number we're repeatedly multiplying by.

  • \( a \) is known as the argument. This is the result of raising the base to a certain power.

  • \( c \) is the logarithm itself. It specifies how many times we multiply the base together to get \( a \).
For example, with \( \log_e(54.598) \approx 4 \), we understand that the base \( e \), when raised to the power of \( 4 \), results in a number approximately equal to 54.598.
This form is particularly useful because it easily translates back to exponential form, allowing calculations to be checked for accuracy and understanding.
Base of Natural Logarithms
The base of natural logarithms is a special mathematical constant, known as \( e \), which is approximately equal to 2.718. It appears often in nature and mathematics due to its unique properties. Some important points about \( e \):
  • It is an irrational number, meaning it cannot be expressed as a simple fraction.

  • It is the base of natural logarithms, often used in growth and decay processes.

  • This constant is entrenched in calculus and the exponential functions, which occur frequently in real-world applications.
Because \( e \) arises in contexts of natural growth like population increase, it is often found in scientific calculations.
Understanding \( e \) helps in applying logarithmic equations to solve complex problems efficiently.
Conversion Between Forms
Converting between logarithmic and exponential forms is a useful skill in mathematics. Here is how the conversion typically works:
  • For any equation in logarithmic form \( \log_b(a) = c \), its corresponding exponential form is \( b^c = a \).

  • This transformation makes it easier to solve the equation or verify solutions.
In practice, to convert \( \log_e(54.598) \approx 4 \) into exponential form, we switch to \( e^4 = 54.598 \).
This conversion is not only about changing the form of the equation; it enables one to calculate or estimate values more directly by utilizing the exponential function.
Verifying these conversions is crucial to maintaining mathematical integrity, ensuring that both forms describe the same relationship between the numbers.

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Most popular questions from this chapter

The radioactive element sodium-24 has a half-life of 15 hr and is used to help locate obstructions in blood flow. If the procedure requires \(0.75 \mathrm{g}\) and is scheduled to take place in 2 days \((48 \mathrm{hr}),\) what minimum amount must be on hand now (to the nearest hundredth of a \(\operatorname{gram})^{2}\)

The radioactive element americium-241 has a half-life of 432 yr and although extremely small amounts are used (about \(0.0002 \mathrm{g}),\) it is the most vital component of standard household smoke detectors. How many years will it take a 10 -g mass of americium- 241 to decay to \(2.7 \mathrm{g} ?\)

Solve each equation by applying fundamental properties. Round to thousandths. $$\log x=1.6$$

Solve each equation using the uniqueness property of logarithms. $$\log _{3}(x+6)-\log _{3} x=\log _{3} 5$$

Use the properties of logarithms to write the following expressions as a sum or difference of simple logarithmic terms. $$\ln (x \sqrt[4]{y})$$
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