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Chapter 4: Problem 15

Solve for the indicated variable. \(M=8.8+5.1 \log D\) for \(D\) (used in astronomy)

Short Answer

Expert verified
\( D = 10^{\frac{M - 8.8}{5.1}} \)

Step by step solution

01

- Isolate the logarithmic term

To solve for the variable \( D \), isolate the term involving \( \log D \). Subtract 8.8 from both sides of the equation:\[ M - 8.8 = 5.1 \log D \]
02

- Divide by the coefficient

Divide both sides of the equation by 5.1 to isolate the logarithm:\[ \frac{M - 8.8}{5.1} = \log D \]
03

- Exponentiate both sides

To solve for \( D \), exponentiate both sides of the equation to get rid of the logarithm. This means raising 10 to the power of both sides (since we assume the logarithm is base 10):\[ D = 10^{\frac{M - 8.8}{5.1}} \]

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Key Concepts

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

isolate logarithmic term
Let's start by isolating the logarithmic term in the equation. The goal is to have the logarithmic term \( \log D \) by itself on one side of the equation. In our exercise, the equation given is: \[ M = 8.8 + 5.1 \log D \] To isolate \( \log D \), we need to eliminate the constant (8.8) on the right-hand side. We do this by subtracting 8.8 from both sides:
\[ M - 8.8 = 5.1 \log D \] Now, the term containing the logarithm is by itself on one side. This helps us in the next steps as we aim to solve for the variable \( D \).
Isolating the term is pivotal because it simplifies the equation, allowing us to handle the logarithmic function more directly.
exponentiating equations
Exponentiating means raising both sides of the equation to a power to cancel out the logarithmic function. To illustrate, once we have the equation: \[ \frac{M - 8.8}{5.1} = \log D \] We need to remove \( \log D \). Since the logarithm we鈥檙e dealing with is base 10, we exponentiate both sides by raising 10 to the power of each side of the equation:
\[ D = 10^{\left( \frac{M - 8.8}{5.1} \right)} \]
Remember these steps when exponentiating:
  • Determine the base of the logarithm (commonly 10 if \( \log \) without any base is mentioned).
  • Raise the base (10) to the power of the entire expression on both sides.
This removes the logarithmic function and simplifies it, giving us the value of \( D \).
logarithmic functions in astronomy
Logarithmic functions are incredibly useful in astronomy for dealing with very large numbers and scales. The given exercise is actually an example of such applications.
For instance:
  • The variable \( M \) often represents the magnitude of celestial objects, like stars.
  • The variable \( D \) often refers to the distance to the object in space.
Using logarithmic functions allows astronomers to compress large ranges of data into simple, manageable figures. For example, the magnitude of stars varies dramatically, and logarithmic scales can express these variations clearly and concisely.
Key benefits of using logarithms in astronomy include:
  • Enabling comparisons between vastly different numbers.
  • Simplifying multiplication and division into addition and subtraction, making calculations easier.
Appreciating the role of logarithmic functions can help in understanding how astronomers analyze and interpret the vast universe.

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Most popular questions from this chapter

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{6}(7 x-2)=1+\log _{6}(x+5)\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}-9 e^{x}-22=0\)

a. The populations of two countries are given for January 1,2000 , and for January 1,2010 . Write a function of the form \(P(t)=P_{0} e^{k t}\) to model each population \(P(t)\) (in millions) \(t\) years after January 1, 2000.$$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Population } \\ \text { in 2000 } \\ \text { (millions) } \end{array} & \begin{array}{c} \text { Population } \\ \text { in 2010 } \\ \text { (millions) } \end{array} & \boldsymbol{P}(t)=\boldsymbol{P}_{0} e^{k t} \\ \hline \text { Switzerland } & 7.3 & 7.8 & \\ \hline \text { Israel } & 6.7 & 7.7 & \\ \hline \end{array}$$ b. Use the models from part (a) to predict the population on January \(1,2020,\) for each country. Round to the nearest hundred thousand. c. Israel had fewer people than Switzerland in the year 2000 , yet from the result of part (b), Israel will have more people in the year \(2020 ?\) Why? d. Use the models from part (a) to predict the year during which each population will reach 10 million if this trend continues.

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{5} 3 $$

Write \(\ln (x+4)=6\) in exponential form.
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