Say you're asked to solve Log 4 16 = X or… without a calculator. But first in this leah4sci.com/mcat
video, we need to understand what the log equations are all about. The typical log equation
is written as Log base B X = Y which simply means B to the Y = X, say what? Google says
that logs are inverse exponents, again, what? Luckily, we don't care what it means so long
as we can solve it quickly right? When you see a log equation, here's what I want you
to do, grab that base and pull it down as your number, grab the answer and pull it down
as your exponent and that is equal to this number right here. In other words, what this
problem is actually asking you is very simple. What is X when the base ten is raised to the
power of the answer, in this case 3? Our new and simplified problem is ten to the third
equals X.
If we go back to the original question, we really have three to the power three is equal
to X, 3 x 3 is 9 x 3 is 27 and so X = 27 and for this one we have 15 to the power of 1
equals x. Well, any number raised to the first is that number and so X is equal to 15 . For
the problems where X is given as the product we approach the same way, ask yourself 4 raised
to the X power or what power is equal to 16 and you should recognize that 4 x 4 or 4 squared
is equal to 16 therefore, X is equal to 2 and here we have 5 raised to the X power is
equal to 125 meaning 5 to what power gives me 125, 5 x 5 is 25 and 25 x 5 is 125 so 5
to the third is equal to 125 giving me that X is equal to 3. Simple enough right? What
happens if the log base is not given? It is not uncommon to see problems like this on
the mcat where you have log of X is equal to 2 but there’s no base. When the base
is not given on the mcat you assume base 10 and treat the problem the same way, we pull
down the base and so we ask 10 to the power of the answer is equal to X, ten to the second
is equal to X. Well, ten to the second is just the scientific notation for the number
100 and so X is equal to 100 or since it's the mcat you can probably just say that X
is equal 10 to the second, same thing.
What about log of X is equal to 7? Ask yourself, ten to the power of 7 is equal to what? Well,
X is equal 10 to the seven. That's my answer there, but if I have to spell it out just
put your 1 and count the zeroes. One, two, three, four, five, six, seven equals X, are
you with me so far? If yes make sure to give this video a thumbs up and let's continue
with the question what if I flip it? What if I ask you to solve log of a thousand is
equal to X? Same thing. Imaginary base 10 to the power of X is equal to 1000, 10 to
what power is equal to a thousand and this is where we use the decimal trick to say one,
two, three, that’s 10 to the power of 3 but the answer is not 10 to the third, X is
simply equal to the exponent that means X is equal to 3. A faster way to do that if
10 to the X is equal to a thousand count your zeroes and that is your answer because isn't
a thousand just 10 to the third? By this logic, if I tell you to solve for a log of one, two,
three, four, five, six, seven, eight, nine zeros is equal to X, all you have to do is
ask yourself, how many zeroes do I have? 9, so if 10 to the power of X is equal to that
whole mess then X is just equal to the number of zeroes and that is 9.
So let's mix it up, if log of 10, meaning 10 to the power of X is equal to 10 that would
be 1, because any number raised to the 1 is equal to itself, what if I ask you for log
of 0.1? Treat it the same way, if in this case we said that 10 to the power of X is
equal to 10 and then we ask 10 raised to what power gives me 10 and that was 1, same thing.
10 to the power of X is equal to 0.1. But how do we solve for this with an exponent
and a decimal? The easiest way is to ask yourself how do I turn that decimal into an exponent,
meaning how do I turn that into scientific notation? I explain this more in the scientific
notation video link below but 0.1 is really just 1 decimal space and that means in scientific
notation, 0.1 is just 1 x 10 to the minus 1 or simply 10 to the minus 1. Now look at
what we did, if 10 to the X is equal to 10 to the minus 1, then isn't X just minus 1?
When a number is raised to a power and it equals the same number raised to a power,
that power has to be the same because it's saying the exact same thing and so 10 to the
X is equal to 10 to the minus 1. Let's try it again, log of 0.001 we have one, two, three
decimal spaces, that means in scientific notation 0.001 is equal to 1 x 10 to the minus 3 or
simply 10 to the minus 3 and so log base 10 of 10 to the minus 3 is equal to minus 3,
just take that power.
Here's how I like to think of it, when you're dealing with large or small numbers where
a large number is greater than 1 for example ten, a thousand, a hundred thousand, a million,
we simply count the zeroes. Remember this problem? We just counted the zeroes, we had
9 zeroes and the answer was 9, if we're dealing with a number less than 1, for example 0.1,
0.001, we simply count the decimal spaces. Just look back at this problem we had 3 decimal
spaces and we had -3. As for the sign, if it's greater than 1, it's more positive so
you get a positive sign, if it's less than 1 it's more negative and so you get a negative
sign, why does this make sense? The way I think of it is we're really just counting
decimals to the right and to the left because if it's greater than 1 aren't zeroes just
decimals in the positive direction and in this case to make a number small, the zeroes
are just decimals in the negative direction, that's how we get our plus and minus and that
is how you can quickly solve the problems without having to think about it too hard.
What if your log problem doesn't have a perfect zero? We have 2 scenarios, if the number is
close to 0 then we just round. Log of 0.12 is about the same as log of 0.1 all we do
is we count the decimal spaces now it looks like I have a decimal space to the right but
the right gives me a whole number to go from 1 to 0.1 that is what I mean by a decimal
space to the left, that’s 1 decimal space in the smaller direction and so I get -1.
On the calculator I get -0.92 which is close enough to -1. If you're asked to solve for
the log of 1094357 just round that as follows and then count your zeroes, one, two, three,
four, five, six, six in the bigger or positive direction gives me positive 6. The calculator
gives me 6.03 which is close enough, if the number is close enough to 1 just round it
down so you have a 1 and zeroes. If the number is really high meaning an 8 or a 9 just round
it up to the next 10, log of 93 is close enough to log of 100, 100 has 2 zeros in the positive
direction which equals 2 and the calculator gives me 1.97 which is close enough. The log
of 0.00945 is close enough to the log of, and this is where you wanna be careful, we
don't turn the 9 into a 1, we turn the 9 into a 10, that means the 1 comes 1 decimal space
to the left, 1 tenths place bigger but the zeros after the 1 don't count. And so we're
really just looking at log of 0.01 which is 2 decimal places in the smaller direction
and so we get -2. The calculator gives me -2.02 which is close enough. Now that we have
basics what if you're asked a question like this, find the pH of a solution that has a
hydronium concentration of 0.0065, in other words, our acid has a molarity of 0.0065 if
you recall from my acid base series at leah4sci.com/mcat-acid-base link below you'll remember that the equation
for pH is equal to -log of the H+ concentration and so in this case what we're solving for
is pH which is equal to X, is equal to the -log of 0.0065 . The problem here is two-fold,
one, we're not dealing with a regular log, this is negative and second, 6 is not a very
clean number, it's not close enough to 1 to round down, it's not close enough to 9 to
round up and even though on the mcat close enough is good enough, this takes too much
rounding.
We need a more accurate method which is exactly what we'll cover in the next video which you
could find linked below along with the entire mcat math without a calculator series, practice
quiz and cheat sheet at leah4sci.com/mcatmath, the link again, leah4sci.com/mcatmath.