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Solving Logarithmic Equations using Exponents

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Today we're going to talk about solving logarithmic equations. So we're going to do a couple of examples. And we'll show you two different methods that you can use to solve logarithmic equations.

So let's start by reviewing some of our logarithmic relationships. The first one is the conversion between an equation in exponential form to logarithmic form. So I have an equation in exponential form, y is equal to b to the x. And I can convert that into logarithmic form.

So the base in my exponential form stays the base in my logarithmic form. The output of my function in exponential form becomes the input, or the argument, of my function in logarithmic form. And the input of my function in exponential form becomes the output in logarithmic form.

So another way to think about that is the base, the value for the base, will stay the same. Our exponent will become the answer for our logarithm. And the answer in exponential form becomes the argument in logarithmic form.

The second relationship is the power property of logarithms, log base b of x to the n is equal to n times log base b of x. So here we take our exponent, n, and we use that. We move it. And it becomes a constant multiplier for the value of our logarithm.

And finally we have the change of base formula. So if we have a logarithm, log base b of x we can write that using common logarithms so when we don't have a base here with our log, that's implied to be a base of 10. And we don't need to write the 10. So using common logarithms, log base b of x just becomes log of x divided by log of b. And this formula is then useful because on your calculator we only have a button for the common logarithm, log base 10.

So let's start by solving a pretty easy logarithmic equation, log base 3 of x is equal to 2. So I'm going to solve this by writing it in exponential form. So in exponential form, I know that my base will stay the same, so the base will be 3. I know that my exponent is going to be the output value in log form, so 3 squared to the second power. And that's going to equal x.

So now that I have this in exponential form, I can simply simplify to find my value for x. 3 squared is equal to 9, so x equals 9.

So for our second example, we've got the logarithmic equation, log base 4 of 5x minus 6 is equal to 3. We could solve this using the method from the first example of writing it in exponential form and then solving for x. However, I'm going to show you using a different method how we can solve this equation.

So we know that exponentials and logarithms are inverses of each other, so we can use those operations to undo each other. So if I have a logarithmic logarithm operation, I can use the exponential operation to cancel it out. So I can raise both sides of my equations. So I can do 4 to the exponent of the entire side, each side of my equation.

So each side of my equation is going to become the exponent. And I'm going to use 4 as the base, which came from this value here. So what I have now is 4 to the value of log base 4 of 5x minus 6 is equal to 4 to the third.

So it looks a little bit strange. But the reason we do that is because now these two operations will cancel out. 4 to the exponent cancels out log base 4. So with these two operations canceled out, we simply on this side have 5x minus 6 is equal to-- and 4 to the third power where is the 4 times 4 times 4, which is 64.

So now this equation looks much more simple to solve. To isolate my x variable, I'll start by adding 6 on both sides. So this gives me 5x is equal to 70. And then I'll divide by 5 on both sides to get x by itself. And I find that x is equal to 14.

So let's go over our key points from today. Use properties of logarithms to solve logarithmic equations, the power property, change of base formula, and conversion between logarithmic form and exponential form. One method of solving logarithmic equations involves converting the equation from logarithmic to exponential form. And a second method involves writing an exponential on both sides of the equation, canceling out the logarithm operation, and solving the resulting equation.

So I hope that these key points and examples helped you understand a little bit more about solving logarithmic equations. Keep using your notes. And keep on practicing. And soon you'll be a pro. Thanks for watching.