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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

Solve each equation and check your answers. $$\log (2 x-5)-\log 78=-1$$

The growth of a bacteria population: \(P(t)=1000 \cdot 3^{t}\) If the initial population of a common bacterium is 1000 and the population triples every day, its population is given by the formula shown, where \(P(t)\) is the total population after \(t\) days. (a) Find the total population \(12 \mathrm{hr}, 1\) day, \(1 \frac{1}{2}\) days, and 2 days later. (b) Do the outputs show the population is tripling every \(24 \mathrm{hr}\) ( 1 day)? (c) Explain why this is an increasing function. (d) Graph the function using an appropriate scale.

For simple interest accounts, the interest earned or due depends on the principal \(p\), interest rate \(r\), and the time \(t\) in years according to the formula \(I=p r t.\) Find \(p\) given \(I=\$ 229.50, r=6.25 \%,\) and \(t=9\) months.

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

Graph each function \(f(x)\) and its inverse \(f^{-1}(x)\) on the same grid and "dash-in" the line \(y=x\). Note how the graphs are related. Then verify the "inverse function" relationship using a composition. $$f(x)=\sqrt[3]{x-7} ; f^{-1}(x)=x^{3}+7$$
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