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Use the change of base formula to find log base 5 of 100,

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to the nearest thousandth.

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So the change of the base formula is a useful formula,

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especially when you're going to use a calculator,

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because most calculators don't allow you to arbitrarily change

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the base of your logarithm.

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They have functions for log base e,

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which is a natural logarithm.

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And log base 10.

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So you generally need to change your base.

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And that's what change of base formula is.

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And if we have time, I'll tell you why it makes a lot of sense

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or how we can derive it.

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So the change of the base formula just tells us that

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-- and let me do some colors here --

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log of base a of b is the same thing as log base x,

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where x is arbitrary base of b over log base x over a.

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The reason why this is useful is that we can change our base.

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Here our base is a and we can change it to base x.

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So if our calculator has a certain base x function

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we can convert to that base, it's usually e or base 10.

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Base 10 is an easy way to go.

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And in general if you just see someone write a logarithm like this.

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If they just write log of x -- they're implying, this implies

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log base 10 of x.

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If someone writes natural log of x,

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they're implying log base e of x.

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And e is obviously the number 2.71...keeps going on and on forever.

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Now, let's apply it to this problem.

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We have logarithm -- I'll use colors -- base 5 of 100.

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So this property, this change of base formula tells us that

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this is exact same thing as log -- I'll make x = 10 --

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log base 10 of 100 divided by log base 10 of 5.

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And actually we don't even need a calculator to evaluate this top part.

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log base 10 of 100 -- what power I have to rise 10 to get to 100?

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The second power.

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So this enumerator is just equal to 2.

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So this simplifies to 2 over log base 10 of 5.

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We can now use our calculator, because log function

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on the calculator is log base 10.

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So let's get our calculator out. And we're going to get,

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we want -- let me clear this -- 2 divided by,

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when someone just writes log, they mean base 10.

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They press ln, they mean base e. So log without any other information

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is log base 10.

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So it is log base 10 of 5 is equal to,

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and they want us to round to the nearest thousandth,

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so 2.861.

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So this is approximately equal to 2.861.

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And we can verify it, because in theory if I raise 5

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to this power I should get 100. And it kinda makes sense,

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cause 5 to the 2nd power is 25, 5 to the 3rd power is 125.

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And this is in between the two and it's closer to the third power

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than it is to the second power and this number is closer to three

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than it is to 2.

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But let's verify it, so if I take 5 to that power.

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And let me type in what we did, to the nearest thousadth.

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5 to the 2.861, so I'm not putting in all of the digits.

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What do I get?

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I get 99.94. if I'd put all of those digits in, it should get pretty close

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to 100.

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So that's what make you feel good that this is the power

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you have to raise 5 to get to 100.

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Now that out of the way, let's actually think about

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why this property, why this thing right over here makes sense?

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So if I write log base a, I'm trying to be fair to the colors,

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log base a of b, let's say I said that to be equal to

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some number, let's call that's equal to c.

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So that means a to c-th power is equal to b.

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This is an exponential way of writing this truth.

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This is a logarithmic way of writing this truth.

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... is equal to b.

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Now we can take the logarithm of any base of both sides of this.

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Anything you do... if you say 10 to the what power is equal this.

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10 to the same power will be equal to this,

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cause those two things are equal to each other.

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So let's take the same logarithm of both sides of this.

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So logarithm with the same base.

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And I actually will do log base x, to prove the general case here.

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So I'm going to take log base x of both sides of this.

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So this is log base x of a to the c-th power.

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I'm trying to be faithful to the colors.

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Is equal to log base x of b.

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Let me close it of with orange as well.

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And we know from out logarithm properties:

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log of a to the c is the same thing as c times the logarithm

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of whatever base we are of a.

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And of course this going to be equal to log base x of b.

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Let me put b right over there.

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and if we want to solve for the c,

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you just divide both sides by log base x of a.

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So you get c is equal to and I'll stick to the color --

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so it's log base x of b over log base x of a.

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And this is what c was, c was logarithm base a of b.

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... is equal to log base a of b.

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Let me write it... let me do it in the original colors,

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just so it becomes very clear what I'm doing.

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I think you know where this is going.

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But I want to be fair to the colors.

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So c is equal to log base x of b over --

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let me scroll down a little bit --

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over log base x -- dividing both sides by that -- of a.

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And we know from here, I can just copy and paste it.

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This is also equal to c.

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This is how we defined it.

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So let me copy it and then we paste it.

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This is also equal to c.

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And we're done.

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We've proven the change of the base formula.

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Log of base a of b is equal to log base x of b divided by log base x of a.

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And in this example: a was 5, b is 100

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and base we switched to is 10 -- x is 10.