Base e for Natural Logarithms Problem: Figure 6-3g shows the graph of y = ln x and the

Differentiate: y = tan1 x

Integrate: (4x + 1)1 dx

Find: lim (sin x)/(x)

log 12 log 48 = log ?

log 7 + log 5 = log ?

3 log 2 = log ?

Sketch the graph of y dx for the function shown in Figure 6-3e. Figure 6-3e

Write one hypothesis of the mean value theorem

Write the other hypothesis of the mean value theorem

If y = x3, then the value of dy when x = 2 and dx = 0.1 is ?. A. 12 B. 1.2 C. 0.8 D. 0.6 E. 0.4

For Problems 16, show that the properties of ln are true by evaluating both sides of the given equation.ln 24 = ln 6 + ln 4

For Problems 16, show that the properties of ln are true by evaluating both sides of the given equation. ln 35 = ln 5 + ln 7

For Problems 16, show that the properties of ln are true by evaluating both sides of the given equation. ln = ln 2001 ln 667

For Problems 16, show that the properties of ln are true by evaluating both sides of the given equation.ln = ln 1001 ln 77

For Problems 16, show that the properties of ln are true by evaluating both sides of the given equation.ln (17763) = 3 ln 1776

For Problems 16, show that the properties of ln are true by evaluating both sides of the given equation. ln (10664) = 4 ln 1066

For Problems 7 and 8, prove the properties. Do this without looking at the proofs in the text. If you get stuck, look at the text proof just long enough to get going again.Prove the uniqueness theorem for derivatives.

Prove that ln x = loge x for all x 0

Prove that ln (a/b) = ln a ln b for all a 0 and b 0.

Prove that ln (ab) = b ln a for all a 0 and for all b.

Prove that ln (a/b) = ln a ln b again, using the property given in Problem 10 as a lemma.

Prove by counterexample that ln (a + b) does not equal ln a + ln b.

Write the definition of ln x as a definite integral.

Write the change-of-base property for logarithms.

For Problems 15 and 16, find an equation for the derivative of the given function and show numerically or graphically that the equation gives a reasonable value for the derivative at the given value of x.f(x) = log3 x, x = 5

For Problems 15 and 16, find an equation for the derivative of the given function and show numerically or graphically that the equation gives a reasonable value for the derivative at the given value of x. f(x) = log0.8 x, x = 4

For Problems 1724, find an equation for the derivative of the given function.g(x) = 8 ln (x5)

For Problems 1724, find an equation for the derivative of the given function.h(x) = 10 ln (x0.4)

For Problems 1724, find an equation for the derivative of the given function.T(x) = log5 (sin x)

For Problems 1724, find an equation for the derivative of the given function.R(x) = log4 (sec x)

For Problems 1724, find an equation for the derivative of the given function. p(x) = (ln x)(log5 x)

For Problems 1724, find an equation for the derivative of the given function.q(x) =

For Problems 1724, find an equation for the derivative of the given function.f(x) = ln

For Problems 1724, find an equation for the derivative of the given function.f(x) = ln (x4 tan x)

Find (ln x3x

Find (ln 5sec x)

Lava Flow Problem: Velocities are measured in miles per hour. (When a velocity is very low, people sometimes prefer to think of how many hours it takes for the lava to flow a mile.) Lava flowing down the side of a volcano flows more slowly as it cools. Assume that the distance, y, in miles, from the crater to the tip of the flowing lava is given by y = 7 (2 0.9x) here x is the number of hours since the lava started flowing (Figure 6-3f). Volcanologists in Hawaii stand on the fresh crust of recently hardened lava. a. Find an equation for dy/dx. Use the equation to find out how fast the tip of the lava is moving when x = 0, 1, 5, and 10 h. Is the lava speeding up or slowing down as time passes? b. Transform the equation y = 7 (2 0.9x) (from part a) so that x is in terms of y by taking the log of both sides, using some appropriate base for the logs. c. Differentiate the equation given in part b with respect to y to find an equation for dx/dy. Calculate dx/dy when y = 10 mi. What are the units of dx/dy? d. Calculate dx/dy for the value of y when x = 10 h. e. You might think that dy/dx and dx/dy are reciprocals of each other. Based on your answers above, in what way is this reasoning true and in what way is it not true?

Compound Interest Problem: If interest on a savings account is compounded continuously, and the interest is such that the annual percentage rate (APR) equal 6%, then M, the amount of money after t years, is given by the exponential function M = 1000 1.06t . You can solve this equation for t in terms of M. a. Show the transformations needed to get t in terms of M. b. Write an equation for dt/dM. c. Evaluate dt/dM when M = 1000. What are the units of dt/dM? What real-world quantity does dt/dM represent? d. Does dt/dM increase or decrease as M increases? How do you interpret the real-world meaning of your answer?

Base e for Natural Logarithms Problem: Figure 6-3g shows the graph of y = ln x and the horizontal line y = 1. Because logb b = 1 for any permissible base b, the value of x where the two graphs cross must be the base of the ln function. By finding this intersection graphically, confirm that e is the base of the natural logarithm function. Figure 6-3g

Journal Problem: Update your journal with what youve learned about the natural and base-b logarithmic functions in this section.