Lesson 2 Log In, Log Out Develop Understanding

Ready

Find the answer to each logarithmic equation.

(Note: If the base of the log is missing, as in question 5, it is assumed to be base $10$ .)

1.

$\log_{5} 625 =$

2.

$\log_{3} 243 =$

3.

$\log_{5} 0.2 =$

4.

$\log_{9} 81 =$

5.

$\log 1,000,000 =$

6.

$\log_{2} 32 =$

7.

$\log_{3} 3^{11} =$

8.

$\log_{8} 8^{- 2} =$

9.

$\log_{x} x^{7} =$

10.

Write an exponential equation for the logarithmic equation provided.

$\log_{b} d = c t$

Set

11.

The diagram shows the graphs of $f (x) = 2^{x}$ , $g (x) = 3^{x}$ , and $h (x) = 5^{x}$ .

What point is shared by all three exponential functions?
Why is this true?
Draw a vertical line where $x = 1$ . Label the point where each graph intersects the $x = 1$ line. What do you notice?
Graph the points for $a (1) = 11^{x}$ and $b (1) = 13^{x}$ .
Sketch the graphs of $p (1) = 11^{x}$ and $q (1) = 13^{x}$ .
Any exponential equation in the form of $y = b^{x}$ will contain the two points and .

The graphs of two different logarithmic equations are given. Use your answers from a–d to help you find the equation of each graph. Remember to include your base.

12.

a.

What is the value of $x$ when $f (x) = 0$ ?

b.

What is the value of $x$ when $f (x) = 1$ ?

c.

What is the value of $x$ when $f (x) = 2$ ?

d.

What is the value of $x$ when $f (x) = 4$ ?

e.

What is the equation of the graph?

13.

a.

What is the value of $x$ when $f (x) = 0$ ?

b.

What is the value of $x$ when $f (x) = 1$ ?

c.

What is the value of $x$ when $f (x) = 9$ ?

d.

What is the value of $x$ when $f (x) = 4$ ?

e.

What is the equation of this graph?

Go

Rewrite each expression. Answers should have only positive exponents.

14.

${(2^{3})}^{4}$

15.

${(x^{3})}^{2}$

16.

${(a^{3})}^{- 2}$

17.

${(2^{3} w)}^{4}$

18.

${(b^{- 7})}^{3}$

19.

${(d^{- 3})}^{- 2}$

20.

$x^{2} \cdot {(x^{5})}^{2}$

21.

$m^{- 3} \cdot {(m^{2})}^{3}$

22.

${(x^{5})}^{- 4} \cdot x^{25}$

23.

${(5 a^{3})}^{2}$

24.

${(6^{- 3})}^{2}$

25.

${(2 a^{3} b^{2})}^{2}$