Mathematical Investigations III
Name:
Mathematical Investigations III
Logarithms
The Second Law of Logarithms
On the same coordinate axes, graph the following functions using an appropriate window. All
are base 10.

f1 (x)  log x 3
f2 ( x)  log( 3)  log( x)

f3 (x)  3log x

f 4 ( x)   log  x  
3
Describe the graphs. Which, if any, are the same?
The two identical graphs give us a result that we will call the Second Law of Logarithms.
THE SECOND LAW OF LOGARITHMS:
log b x k  k  log b x , b  0, b  1
 
We must prove this.
As before, we begin by naming the log b x .

Let log b x  p .


Change this into exponential form.
Since we are interested in x k , raise both sides of your equation to the kth power and use a known
property of exponents to simplify.
Finish the proof. (Note: Two quick steps remain. Look back at the last proof if necessary.)
Logs 5.1
Rev. S03
Mathematical Investigations III
Name:
Practice:
1.
Rewrite the following using the Second Law of Logarithms and simplify if possible.
2.

b.
log2 35 =

d.
log3 x 4 =
a.
logb b7 
c.
log3 95 
e.
log 49 73x 

 
 
 

Prove: If m > 0, then log5 m3n  3nlog5 m . Use the method of proof used on page 5.1.
Start by choosing what expressions to define as p and q.
Logs 5.2
Rev. S03