Round-off error can get really big really fast with logs, and you don't want to lose points because you rounded too early and thus too much. In other words, when you plug your decimal approximation into the original equation, you're just making sure that the result is close enough to be reasonable.
: Given the equation of a function, identify a possible graph (among 4) corresponding to the given function.
Algebraic functions including linear, square root, quadratic and absolute value functions are included.
But for the rest of this example, I'll just skip writing the 10 just because it'll save a little bit of time. So this expression, right over here, is the power I have to raise 10 to to get x, the power I have to raise 10 to to get 3.
Now with that out of the way, let's see what logarithm properties we can use.
Allowing for round-off error, these values confirm to me that I've gotten the right answer.
If, on the other hand, my solution had returned a value of, say, Let's do a couple more examples. is the answer, I first need to check (especially because this answer is negative) whether it'll work in the original equation.
An example would be: , is a valid solution, and often this will be all that I'm supposed to give for the answer.
However, in this case (maybe leading up to graphing or word problems) they want me to provide a decimal approximation.
So right over here, we have all the logs are the same base. So by this property right over here, the sum of logarithms with the same base, this is going to be equal to log base 3-- sorry, log base 10-- so I'll just write it here. Then, based on this property right over here, this thing could be rewritten-- so this is going to be equal to-- this thing can be written as log base 10 of 4 to the second power, which is really just 16. And then we still have minus logarithm base 10 of 2.
And now, using this last property, we know we have one logarithm minus another logarithm.