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.

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 🙂

The last example is incorrect it is -3 you messed up on the last line there, thanks for the help

the last equation should be x= -3

This really helped me!!! Thank you so much

Is it possible to do log729 without a calculator?

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?

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

These videos are perfect, they go over exactly what I am learning in pre calc.

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.

Very helpful. Thanks

a math god

This video helped so much, thank you. Especially for national benchmark tests😆

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!

Thanks You!

bless this mans

Damn, he makes it look so easy.

The last one should be -3. Other than that, this video helped me so much. Thank you.

thanks