Master Evaluating a logarithm without a calculator

Master Evaluating a logarithm without a calculator


Welcome ladies and gentlemen. And what I’d like
to do is show you how to evaluate logarithms
without using a calculator. Now, obviously for a
majority of these– or actually for
all of these, you could type them into
your calculator using the change of base formula
unless your calculator can do different bases. But none of these I
chose base 10 or I didn’t do any natural logarithms. But for all logarithms
you can always just type in your calculator. But I guarantee I
could do all of these as fast as it would take you
to pick up your calculator and plug them in. And it’s just because using
the understanding of logarithms has helped me been able
to do all these problems, as well as negative exponents. So that’s one thing I want
to write into here is– well, there’s a couple things. First of all, remember
we have a logarithms base b of x equals y. We can always rewrite that in
logarithmic form as b to the y equals x. The other thing I
want to remind you is if I have an exponent
in the denominator, I can always rewrite that as
an exponent in the numerator, but just using it
with a negative power. You can put that
over 1 if you wanted to but we don’t really need to. So basically, by knowing
these two understandings, I can do all these problems
without a calculator very, very quickly. The other thing,
actually, we also want to know is using
the one to one property is sometimes helpful even
though we’re not solving, we’re just evaluating. A lot of these, it is very
important to understand this because I think I’m going to
do some difficult problems. So if we have an exponent
equal to another exponent where the bases
are exactly same, then their powers are same. So we can set their powers
equal to each other. OK, now, you can see that
these are just expressions that we want to evaluate. But our basic mode of
thinking is, again, to find what their value is. Now, remember when we’re looking
at a logarithmic equation– for instance, log
base 3 of 9– we know that what that answer
is going to be is 2. Because what a logarithm says
and means is 2 raised to– oh, I’m sorry, 3 raised
to what power is 9? 2, and if you were to rewrite
that in exponential form, it would look like this. 3 squared equals 9. So again, that’s writing it
into my exponential form. 3, the base, raised to
what power gives you 9? That’s what our logarithm
is basically asking us. 3 raised to what power is 9? 2, and you can always
think about rewriting it in exponential form to
kind of check your answer. So, in my first
example I kind of tried to think of one that’s
going to be fairly simple. This is basically stating us
5 raised to what power is 25? Well, I know that 25
is a square number. And I know that 5 squared is 25. So therefore the answer
here in this case, it’s just going to be 2, OK. Now, the next one is 3
raised to what power is 81? Now again, without
a calculator, you need to kind of start knowing
the powers of 3, 4, 5, 6, 2. So what I would start
doing, is if you’re not really familiar what this would
be, start raising 3 to powers. 3 to the first power is 3. 3 squared is 9. 3 cubed you might have to
type in your calculator, but eventually you’ll
understand it’s 27. 3 to the fourth power you might
have to type in your calculator originally, but you’ll
see that it’s 81. So that means 3 raised
to what power is 81? Well, the answer
is equal to 4, OK. Now, in the next example
what I’m going to do is I’m just going
to kind of think about it in another format. Now, you could do the exact
same process I did over there, and just start listing 2 to what
power, you know, equals that. But I also want you to
think about this as maybe possibly an equation. Here’s an expression. Now, again these
are expressions. We’re just evaluating. We’re not equations. But really what an
expression is is you’re saying it’s
equal to what value? So I’m actually going
to replace that as an x. And then what I
could do is using, by rewriting this
as exponential form, I can rewrite this as
2 to the x equals 64. So therefore again, if you’re
having trouble understanding what the logarithm is,
set it equal to x and then rewrite an exponential form. 2 raised to what power is 64? And we’ll get to the one to
one property here in a second. But what we could do, again,
is just think about this. Well, if I know that 2 to
the first, 2 the second, 2 to the third, 2 to the fourth,
2 to the fifth, 2 to the sixth. 2, 4, 8, 16, 32, 64. So 2 to the 6th power. So therefore, I know that x,
or that’s equal to, 6, OK. Now, where that variable
kind of comes in handy though is in a
problem like this, 8 raised to what power is 4? Oh, I don’t know 8 raised to 8. 8 raised to the
first power is 8, so that’s already larger than 4. So how would I do this? Again, if you use the method
that I did in the last one, you set it equal to x. Now, rewriting it
in exponential form, gives me 8 to the x equals 4. Well, now I can use this one
to one property that I stated. And by using the one
to one property, what I notice is 8 and 4, they have
to have the same base, right. Well, 8 and 4 do not
have the same base. But you can use
the same base of 2. Because 2 squared is
4 and 2 cubed is 8. So then what I do is I rewrite
this as 2 cubed raised to the x equals 2 squared. Now, that they
have the same base, I can just set the powers
as equal to each other. So I have 3x is equal to
2 so x equals 2/3, OK. All right, so now we’re going
to get into some fractions. Again, into the same
respect, main important thing that I can do is I can
rewrite all fractions– again, you could think of this as 4
to the positive first power– you can think of all
fractions as rewriting them with negative powers. So now I can rewrite
this as log base 4 of 4 to the negative first power. Instead of it being
to the first power, you can write it to the
negative first power. Now, there’s another
rule of logarithms that you should remember. Log base b of b equals 1. And log base b of b
to the x equals x. Well, if I’m taking the log
of the same base of 4, raised to the negative 1 using
this rule of logarithm, my solution is just going
to be a negative 1, OK. This example might be
a little bit confusing. I have fraction and fraction. I could rewrite them
as negative powers. But again, what I’m
seeing here is probably rewriting them in exponential
form would probably make more sense for me. So I’m going to do 1 over 6
raised to the x equals 1/36, OK. Now, I use my negative powers. So I’ll do 6 to the negative
first, raised to the x, equals 36 raised to the x. OK, now I want to see– oops,
that’s negative first power, isn’t it? OK, now I want to see,
can I rewrite 6 and 36 to the same base? Well, yeah, 6, 36 squared
is– 6 squared is 36. So I get 6 to the negative x
when I apply the power rule. Remember, the power
rule states if you have an exponent raised
to another exponent, you multiply them. 36, I can rewrite as 6
squared times negative 1, which is raised to the x. Again, you’re just
going to multiply these. The 6 is divided out using
that one to one property. So I have negative
x equals negative 2. What happened here? Why do I have an x up? I don’t have an x over here? Why is there an x over here? I think I was going crazy. So that equals negative
2 divide by negative 1, divide by negative
1, x equals 2. OK, in this case,
basically again what I’d like to do– OK,
basically in this case is, again, I could use my negative
powers like I did over here. So I can rewrite this
as log base 2 of 8 to the negative first power. Now, again that’s
still not really, you know, helping me out, right? But again, think about
this as using this rule. Can I rewrite 8 as a base 2? Because if I can
rewrite it as a base 2. It’s the base raised
up of the same base. And yes, I can. Log base 2 of 8 can be
written as 2 cubed raised to the negative 1, which is log
base 2 of 2 to the negative 3– because you multiply
those powers. So therefore, my answer is
just equal to negative 3. OK, and last but
not least, I have log of 1/3 raised to the 27th. Again, what I would
do in this case is I would set this equal
to x and I’d rewrite it as an exponential equation. Then I’d use my negative
powers, 3 to the negative 1, x equals 27. Then what I want to
do is I say, can I rewrite 3 raised to–
3 as– can I have 3 and 27 with the same base? And yes, 3 to the
first power is 3. 3 to the third power is 27. So therefore, I just need
to rewrite 27 as 3 cubed. Therefore, negative x
equals 3, x equals 3. So there you go,
ladies and gentlemen. That is how you evaluate
logarithm expressions without using a calculator. Thanks.

18 thoughts to “Master Evaluating a logarithm without a calculator”

  1. Thanks for yet another great video!! I like how you give multiple difficult problems. My exam is tomorrow and you have helped in giving me the confidence to know I can succeed. P.S. I believe the last answer is supposed to be -3 sinced you divided by -1 🙂

  2. Thank you SO MUCH for this video brilliant! Question plz on last seconds of your video, X = MINUS 3, not +3 .. or am i missing something?

  3. Hmm.. I have a little trouble understanding this one Mr. Brian.

    5^(x) = 29

    I tried rewriting this in ligarithmic form log5(29) = x

    that still doesn't give me enough info to solve it. This problem looks most similar to the 4th one you did in the video, but there's no way a 29 and a 5 could share a base. The answer choices on my question all contain ln. I don't know what else to do

  4. I also learned this, think of decimal exponents just simply as products of exponents and roots. Example 2.72^2.25 = x^2+x^0.2+x^0.05 = x^2 * sqrt 1/0,2 times 1/0.05 means 5th and 20th root. 20th root = 10th root and sqrt or 5th root and two times the sqrt. So I do 2.72^2 * 5th root of 7.72 * 20th root of 2.72. So decimals are just the roots.

  5. I have watched tons of your videos and I am sure I speak for many others as well when I say that your videos have really helped get me that extra clarification that I may not be able to immediately get from my teacher, so thank you for that!

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