Key to the Practice Midterm

Key to the Practice Midterm

I. Homework

II. Review For Midterm III

Exponentials

Logs: Definitions and graphs

Properties of logs

Exponential and Log equations

Applications

Sequences

Series

Arithmetic Sequences and Series

Geometric Sequences and Series

The Binomial Theorem

In class, we will go over any questions that the students have.

III. Key For the Practice Midterm III

Problem 1:

A) 2log₂ (x - 1) - log₂ (x² - 1) = 2

log₂ (x - 1)² - log₂ (x² - 1) = 2

log₂ [(x - 1)² /(x² - 1)] = 2

log₂ [(x - 1)² /(x - 1)(x + 1)] = 2

log₂ [(x - 1)/(x + 1)] = 2

(x - 1)/(x + 1) = 4

x - 1 = 4x + 4

-3x = 5

x = -5/3

but since -5/3 - 1 < 0 there is no solution.

B) log 3^{2x + 2}= log 5^{3x + 4}

(2x + 2)log3 = (3x + 4)log5

x(2log3) + 2log3 = x(3log5) + 4log5

x(2log3 - 3log5) = 4log5 - 2log3

x = (4log5 - 2log3)/(2log3 - 3log5)

C) log 50/log 7 = 2.0103

Problem 2

Will be done in class. Notice that the first graph passes through (1,0) and (5,1) while the second passes through (0,1) and (1,5).

Problem 3

A. (7 - 7²) + (8 - 8²) + (9 - 9²) + (10 - 10²)

Problem 4

A) 3

B) -320

Problem 5

A) 3.05

B) 3

Problem 6:

3.18 years

Problem 7:

1 - 8x + 24x² - 32x³ + 16x⁴