### Calculus and Algebra Practice Problems#### 1. Evaluate the expression **ln e**The natural logarithm of $ e $ (where $ e $ is the base of the natural logarithms, approximately equal to 2.71828) is 1. Therefore, $ \ln e = 1 $.#### 2. Find the domain of the function: $ \ln(9 - x^2) $To determine the domain of the function $ \ln(9 - x^2) $, we need the argument of the logarithm (i.e., $ 9 - x^2 $) to be greater than 0:\[ 9 - x^2 > 0 \]\[ 9 > x^2 \]\[ -3 < x < 3 \]Thus, the domain is $ -3 < x < 3 $.#### 3. Use the properties of logarithms to write the expression as a single logarithm: $ \ln(x) - \ln(x^2 - 1) $Using the properties of logarithms, specifically the difference of logarithms property ($ \ln(a) - \ln(b) = \ln(\frac{a}{b}) $):\[ \ln(x) - \ln(x^2 - 1) = \ln\left(\frac{x}{x^2 - 1}\right) \]#### 4. Use the change of base formula and a calculator to find the solution: $ \log_3 7 $ (round to 3 places)The change of base formula for logarithms is given by:\[ \log_b a = \frac{\ln(a)}{\ln(b)} \]Using this formula:\[ \log_3 7 = \frac{\ln(7)}{\ln(3)} \]Using a calculator to compute:\[ \log_3 7 \approx \frac{1.945910}{1.098612} \approx 1.771 \]Thus, $ \log_3 7 \approx 1.771 $ (rounded to 3 decimal places).

Answered: Image /qna-images/answer/8f99614c-0d3c-4fba-a142-376df72d48f4.jpg

Find the domain of the function: In (9 – x2) Use the properties of logarithms to write the expression as a single logarithm: In (x) – In (x² – 1) Use the change of base formula and a calculator to find the solution: log37 (round to 3 places)

Find the domain of the function: In (9 – x2) Use the properties of logarithms to write the expression as a single logarithm: In (x) – In (x² – 1) Use the change of base formula and a calculator to find the solution: log37 (round to 3 places)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.7: Applications

Problem 14EQ

See similar textbooks

Related questions

Q: Solve for æ: log,( – x + 12) = 2 Write log,( – x + 12) = 2 in expone

A: Given that: log4(-x+12)=2

Q: Solve. log,(3x – 4) = log:(x + 2)

A: To solve the given equation

Q: Solve equation: log:(3x – 1) = 3 Select one: O a. x -3 O b. x-3 O c. x-1/3 Od. x-10/3

Q: Log(2 – 5i) Evaluate:

A: Introduction: The logarithm of a complex number can also be calculated. The obtained result is again…

Q: Solve for x, and show all the steps: logg x + log:(x – 12) = 2 %3D

Q: Solve the equation for x: log, (x-5)+log, (x+3) =1

A: Consider the equation: log9x-5+log9x+3=1. Apply the log property logab+logac=logabc…

Q: Solve for x: log, (3x + 10) = 2 Write log, (3x + 10) = 2 in exponential form. Find the value of x. x…

Q: Write as a single logarithm: 3 ln (x + 7) - ln x

A: 3lnx+7-lnx

Q: 3 Expand the logarithmic expression: In- x 2-4

Q: Solve for x: log, 4 + log, (x + 4) = 3 %3D

Q: use the properties of the natural logarithm to: A. expand the logarithmic expression: ln (2x/x+2)…

A: A) ln2xx+2B) 5 lnx-7 lny-8 lnz

Q: Simplify: 10 log (5x+1)

Q: log4 (3x + 4)5 = 15

A: solve logrithm

Q: Solve for : log, (+ 6) = 2 Write log (a+ 6) = 2 in exponential form. Find the value of x.

Q: You Try: Condense 5log(x) - 3log(y) A. log(x®y³) B. log(xy)? C. log(xly®) D. log(5x3y)

Q: Solve for x : log, x = 4

A: Here we use basic logarithmic properties to solve the given equation

Question

### Calculus and Algebra Practice Problems

#### 1. Evaluate the expression **ln e**
The natural logarithm of $ e $ (where $ e $ is the base of the natural logarithms, approximately equal to 2.71828) is 1. Therefore, $ \ln e = 1 $.

#### 2. Find the domain of the function: $ \ln(9 - x^2) $
To determine the domain of the function $ \ln(9 - x^2) $, we need the argument of the logarithm (i.e., $ 9 - x^2 $) to be greater than 0:
\[ 9 - x^2 > 0 \]
\[ 9 > x^2 \]
\[ -3 < x < 3 \]
Thus, the domain is $ -3 < x < 3 $.

#### 3. Use the properties of logarithms to write the expression as a single logarithm: $ \ln(x) - \ln(x^2 - 1) $
Using the properties of logarithms, specifically the difference of logarithms property ($ \ln(a) - \ln(b) = \ln(\frac{a}{b}) $):
\[ \ln(x) - \ln(x^2 - 1) = \ln\left(\frac{x}{x^2 - 1}\right) \]

#### 4. Use the change of base formula and a calculator to find the solution: $ \log_3 7 $ (round to 3 places)
The change of base formula for logarithms is given by:
\[ \log_b a = \frac{\ln(a)}{\ln(b)} \]
Using this formula:
\[ \log_3 7 = \frac{\ln(7)}{\ln(3)} \]
Using a calculator to compute:
\[ \log_3 7 \approx \frac{1.945910}{1.098612} \approx 1.771 \]
Thus, $ \log_3 7 \approx 1.771 $ (rounded to 3 decimal places).

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step by step

Solved in 4 steps with 4 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Transcendental Expression$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions

Logarithmic From Express the equation in exponential form. (a) In (x+1)=2 (b) In (x1)=4
Suppose that $12,000 is invested in a saving account paying 5.6% interest per year. (a)Write the formula for the amount in the account after t years if interest is compounded monthly. (b)Find the amount in the after 3 years if interest is compounded daily. (c)How long will it take for the amount in the account to grow to $20,000 if interest iscompounded continuously?
How does the power rule for logarithm help whensolving logarithms with the form logb(xn) ?
What type (s) of translation (s), if any, affect thedomain ofa logarithmic function?
The number of bacteria in a culture is increasing according to the law of exponential growth. After 1 hour there are 100 bacteria, and after 2 hours there are 200 bacteria. How many bacteria will there be after 3 hours?
Discuss the meaning ofthe natural logarithm. Whatis its relationship to a logarithm with base b, andhow does the notation differ?
Discuss the meaning of the common logarithm.What is its relationship to a logarithm with base b,and how does the notation differ?
The natural logarithmic function f(x)= In (x1) has the ____________ asymptote x =____.
An investment account was opened with aninitial deposit of 9,600 and earns 7.4 interest,compounded continuously. How much will theaccount be worth after 15 years?