1.find the value of Of each of the following

Question

1.find the value of Of each of the following, giving your answer an integer
a.Log₆4+Log₆9

 

2.Let Log₃P=6 and Log₃Q=7
Find Log₃(P/Q) By expanding the logarithm 

 

 

3.Solve the equations below using exponents or logarithms:
i)4^x+2=14

Log₄(2)+Log₄(5x)= 1/2

Answer 1

Sweet, lot's of logarithm problems.  Most can be solved without a calculator but involve applying the properties of logarithms. 

 

The first problem involves applying the addition property

 

log₆4+log₆9=log₆(4⋅9)=log₆36=log₆6²=2

 

The next requires subtraction property

 

log₃(P/Q)=log₃P-log₃Q=6-7=-1

 

The next will require a calculator but first we need to solve for x.  But how do we get x by itself?

 

Lets take the log base 10 (typically just written as 'log') of both sides

 

log(4^x+2)=log(14)

 

Here we can apply the exponent property of logarithms

 

log(4^x+2)=(x+2)log(4)=log(14)

 

now lets solve for x

 

x+2=log(14)/log(4)

x=log(14)/log(4)-2

 

Now we can calculate the logs using most scientific calculators.  You can do that...

 

The last problem starts by using the addition property again:

 

log₄2+log₄5x=1/2

log₄(2⋅5x)=log₄(10x)=1/2

 

To finish solving for x we have to remember that y=a^x and y=log_ax are inverses.  So if we feed each side of the equation into the function f(x)=4^x we'll get...

 

4^log₄(10x)=4^1/2

10x=2

x=1/5

 

Good Luck

 

Cheers

 

-Chris