Applications of Log Rules

25log10log bb

2log5log25log bbb

03.06.2log5log bb

09.03.06.

25log25log bb 5log25log 2

bb

06.25log2 b

12.5log2 b

We are just substituting in for each expression!

105log50log bb

255log105log bb

2log5log5log bbb

03.06.06.

15.

8log3log8

3log bbb

42log3log bb

222log3log bb

2log2log2log3log bbbb

03.03.03.05.

04.

3log20log3

20log bbb

3log54log bb 3log522log bb

3log5log2log2log bbbb

05.06.03.03. 07.

3

1

3

5

1log

5

1log

bb

3

1

3

13

1

5log1log5

1log bbb

5log3

11log

3

1bb

06.3

10

3

102.02.0

What happened to the log 1?

x1logbTo determine what happens to the log 1, we need to rewrite it in exponential form!

1bx b to what power = 1?

1b0 Therefore, x=0.

01log1b0 b This brings up an important result,

log 1 = 0!

Any base the zero power is one (except 0).

We are now going to do the same type of problems with powers of 10. Remember that your computer can do base 10.

100

460log6.4log

We need to keep the expression equal to 4.6. By rewriting it as 460/100, we can now use the log rules and substitute what we know!

100log460log100

460log

Log 460 is equal to b

Log 100 is equal to 2, use calculator

2b

100015.3log3150log

We need to keep the expression equal to 3150. By rewriting it as 3.15*1000, we can now use the log rules and substitute what we know!

1000log15.3log100015.3log 3a

1000

1.37log0371.log 1000log1.37log 3c

2

1

460log460log 460log2

1 b

2

1

3

13 31500log31500log 31500log

3

1 1000015.3log

3

1

10000log15.3log3

1 4

3

1 a

3

4a

Some random log rules:

1. We can not take the log of zero (0) or a negative.

For what values of x is the expression defined?

2log xb

Since we can not take the log of 0 or a negative, we write an inequality to solve for x.

02 x2 2

2x

Thus, x is defined for all values of x > 2.

For what values of x is the expression UNDEFINED?

32log xb

032 x3 3 32 x

32 x

2

3x

Thus, x is undefined for all values of x less than or equal to -3/2.

Some random log rules:

2. The log of one is always zero. log 1 = 0

01log b

I will now rewrite it exponentially to show you why!

10 b

Anything (except 0) raised to the zero power is 1.

3. The inverse of an exponential function is a log function!

Exponential function

Log function

Remember, when taking an inverse, it is the same as reflecting over the line y=x. You switch x & y for both!

Homework

• Page 5#1,3,4,5

• Page 6#11

• Page 7#6,8,11a,c,12,a,c