# Logarithm state the value of x for which the equation is defined

1. May 11, 2015

### Jaco Viljoen

1. The problem statement, all variables and given/known data
Consider the equation:
3logx5+2logx2-log1/x2=3
a)State which values of x for which the equation is defined.
b)Solve the equation for x.
2. Relevant equations

3. The attempt at a solution
3logx5+2logx2-log1/x2=3
=logx53+logx22-log1/x2=3
=logx125+logx4-log1/x2=3
=logx500-log1/x2=3

=logx2=logx2/logx(1/x)change of base
=-logx2

=logx125+logx4+logx2=3
=logx500+logx2=3
=logx1000=3
=x3=1000
x=10

Please check if you don't mind?

Last edited: May 11, 2015
2. May 11, 2015

### SammyS

Staff Emeritus
Not too difficult to check.

3logx5+2logx2-log1/x2=3

becomes:
3log105+2log102-log1/102=3 .​

See if the following is true.
$\displaystyle\ 10^{\displaystyle\left(3\log_{10}(5)+2\log_{10}(2)-\log_{1/10}(2)\right)}=10^3\$​

3. May 11, 2015

### Jaco Viljoen

Yes it works.
I dont understand what 9.1 wants me to do?

Last edited: May 11, 2015
4. May 11, 2015

### SammyS

Staff Emeritus
What's 9.1 ?

5. May 11, 2015

### Jaco Viljoen

State which values of x for which the equation is defined.

6. May 11, 2015

### Staff: Mentor

For what values of x does logx(something) make sense? Same question for log1/x(something).

7. May 11, 2015

### SammyS

Staff Emeritus
Several ways to figure this out.

1. From definition of the logarithm.
What does it mean, particularly for base, b, if logb(A) = C ?​

2. From change of base, along with knowing the domain of the logarithm function.
What are logx(A) and log1/x(A) ?​
...

8. May 11, 2015

### Jaco Viljoen

logb(A) = C
bC=A
x>1

9. May 11, 2015

### SammyS

Staff Emeritus
Doesn't that mean that 1/x < 1 ?

10. May 11, 2015

### Jaco Viljoen

x>=3

11. May 11, 2015

### BvU

Still lets 1/x < 1 ! But: is that a problem ?

Try to put SammyS' question 1 in words.

12. May 11, 2015

### SammyS

Staff Emeritus
What are the restrictions on the base, b, if C and A are to be real numbers?

I think the change of base route might lead to the answer more quickly, but if you're to understand logarithmic & exponential functions, then eventually you need to confront this issue regarding the base.

13. May 11, 2015

### Jaco Viljoen

a logarithm is undefined when x<0
1/x will be 1/3 but still x>0 so the logarithm will be defined.
x>0

14. May 11, 2015

### SammyS

Staff Emeritus
I don't see what 1/3 has to do with anything here.

Yes, as you state, "a logarithm is undefined when x<0." .... and, yes, "x > 0" .

Now, are you referring to the base of the logarithm ?

15. May 11, 2015

### Jaco Viljoen

Yes, the base of log cannot be smaller than 0

16. May 11, 2015

### SammyS

Staff Emeritus
Can it be zero ?

Why or why not?

17. May 11, 2015

### Jaco Viljoen

It is not defined, cannot be calculated, gives an error.
Its a rule

18. May 11, 2015

### SammyS

Staff Emeritus
What is not defined?

19. May 11, 2015

### Jaco Viljoen

the logarithm

20. May 11, 2015

### SammyS

Staff Emeritus