problem with logs with exponent

[allegansveritatem]

I am having trouble understanding how to solve log equations with exponents. Here is a problem I tried to solve. I came up with a wholly different solution than the book I use did. But I can't understand why my way is wrong . Here is the way I did it:


and here is the way it was done by Earl Stokowski:



Why didn't my way work?

 

[Dr.Peterson]

There's a sign error going from the 4th line to the 5th in your work, and another within the 5th line.

I prefer to avoid decimals until the end, as in the provided solution; but if you get the arithmetic right, your work should be good.

 

[allegansveritatem]

There's a sign error going from the 4th line to the 5th in your work, and another within the 5th line.

I prefer to avoid decimals until the end, as in the provided solution; but if you get the arithmetic right, your work should be good.

well, reading over my OP I notice that when I copied the photo I had made of the book solution I left out the actual number which is: -3.64. This is how I know I did something very wrong but the solution I post here is the second one I did and I got the same thing both times.. But, are you saying that the procedure I followed could produce a correct answer provided my arithmetic were good?

 

[Steven G]

Your main mistake is thinking that \(\displaystyle \dfrac {\log5}{\log6} = 1.11328\) which is not true.

 

[Dr.Peterson]

Your main mistake is thinking that \(\displaystyle \dfrac {\log5}{\log6} = 1.11328\) which is not true.

It's [MATH]\frac{\log 6}{\log 5} = 1.11328[/MATH], and it is true.

well, reading over my OP I notice that when I copied the photo I had made of the book solution I left out the actual number which is: -3.64. This is how I know I did something very wrong but the solution I post here is the second one I did and I got the same thing both times.. But, are you saying that the procedure I followed could produce a correct answer provided my arithmetic were good?

The sign errors are huge. FIX THEM, and then write back! Until then, there's no need to ask us about it.

 

[allegansveritatem]

Your main mistake is thinking that \(\displaystyle \dfrac {\log5}{\log6} = 1.11328\) which is not true.

I will go over the whole problem using the same procedure again today and get back to this thread tonight. Thanks for the tip.