Hey guys just found this site and looks awesome. Unfortunately my math teacher does not explain a lot of things in much detail. I am working on our exam practice test and this is one of the questions... Solve for x in each equation. Provide an exact answer and an approximate answer. (a) log base 4 (3x+7)=2 . I did b^x=y and solved for x. So x=3. Is this correct and is this an approximate or exact answer? I also put that into the equation so you get log base 4 (16) =2. So I am confused on which is the exact and approximate answers. Thanks!
Question help.. what is the exact and approximate answer?
- Thread starter jakebake
- Start date
pka
Elite Member
- Joined
- Jan 29, 2005
- Messages
- 11,989
Yes that is correct. As to to being exact or approximate, it depends on who is asking the question.
If \(\displaystyle \log_b(ax+c)=d\) then if \(\displaystyle b>0~\&~a\ne 0\) then \(\displaystyle x=\dfrac{b^d-c}{a}\) is an exact solution.
On the other hand, if we have \(\displaystyle \log_{10}(2x+1)=.5\) then \(\displaystyle x=\dfrac{\sqrt{10}-1}{2}\) is exact but not in the ways some instructors mean the term.
For me, "exact" is what you get by solving algebraically, and
............"approximate" is what you get solving numerically or using a calculator.
For instance, if you find a root of a polynomial by factoring, it is exact,
but if you find the root using Newton's method on a computer, it is approximate.
If your result includes an irrational number like the square root of 10, then
\(\displaystyle \sqrt{10}\) is exact, and \(\displaystyle 3.16227766016\) is approximate.
Your solutions to a and b are both exact. Given that you have an exact answer, it seems odd to try to find an approximate answer! Perhaps if you converted to natural logarithms, then you would have to use your calculator to get the result, so that would be approximate.
HallsofIvy
Elite Member
- Joined
- Jan 27, 2012
- Messages
- 7,763
Was that the only equation or were there a list of them with the same instruction?
If you have \(\displaystyle log_4(3x+7)= b\) then \(\displaystyle 3x+ 7= 4^b\). As long as "b" is an integer, we can solve for the exact \(\displaystyle x= \frac{4^b- 7}{3}\). If b is NOT an integer, then x cannot be written in "exact" form, except as the formula given, and you can write it only approximately as a decimal number.
For example, if b= 3, we could write the "exact" solution as \(\displaystyle x= \frac{\sqrt[3]{4}- 7}{3}\) or approximately as -1.80420.
However, the problem, as given, has the exact solution x= 3 and there is no need to write an "approximate" solution.
...