Solving for a in Logarithmic Equation: x(x-15)=2

[goosey00]

In log x + (x-15)=2 and I have to convert it to an exponental. On the practice one they said in the form of log(small)M+log(small)aN=log(small)aM*N
so log(small)ax a^x=x
a=10
Why in this equation does a=10?? I don't understand. In the equation above I understand it being x(x-15)=2 just not the a=10 part Please help

 

[Jameson]

Re: log./^ help

In log x + (x-15)=2 and I have to convert it to an exponental. On the practice one they said in the form of log(small)M+log(small)aN=log(small)aM*N
so log(small)ax a^x=x
a=10
Why in this equation does a=10?? I don't understand. In the equation above I understand it being x(x-15)=2 just not the a=10 part Please help

Hi goosey00,

Should there be another log? You wrote "log x + (x-15)=2" but I think it might be "log x + log(x-15)=2"

 

[goosey00]

your right. BTW-the other day when we had that long thread, my teacher said I didn't even have to go as far as I did. All that time we spent-really? Oh well.

 

[Jameson]

Ok, so we have \(\displaystyle \log x + \log (x-15)=2\). A rule you need to know, among a couple of others, is \(\displaystyle \log a +\log b = \log (ab)\). You can go the other way as well. So combine the two log expressions into one and then use the definition of log to rewrite the equation. What do you get for that?

 

[goosey00]

Im sorry I am still confused. The end is -2(a*b) right. ??

 

[Jameson]

\(\displaystyle \log x + \log (x-15)=2\)

\(\displaystyle \log[x*(x-15)]=2\)

Assuming log x means base 10, then \(\displaystyle 10^2=x(x-15)\)

Can you go further?

 

[goosey00]

SO, is the rule of a base is 10. You wrote assumed. Thats the confusing part is the 10. I can solve it from there. I just am missing something.

 

[Jameson]

SO, is the rule of a base is 10. You wrote assumed. Thats the confusing part is the 10. I can solve it from there. I just am missing something.

The way logarithms are defined is \(\displaystyle \log_{a}b=x \implies a^x=b\) When there is nothing written in subscript then we can assume it's 10 (in higher math it could mean "e"). For this problem I think it's safe to say the base is 10. However, you need to remember the definition above in order to switch between the log form and exponential form of an expression.

Did your teacher explain what logarithms are and how you can use certain properties to manipulate them?

 

[goosey00]

not really but what you just said completely makes sense now. Thanks again. Jenny

 

[Jameson]

not really but what you just said completely makes sense now. Thanks again. Jenny

The way to learn logs is to make sure you know and understand the definition that I gave you in my last post. It's just a way to rewrite something in an easier form. So be comfortable writing exponents to logs and logs to exponents. Then you'll be introduced to 3-4 rules that only apply to logs and almost all of the problems you'll see use them. Practice, practice, practice.