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Solving Exponential Equations using Logarithms

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Today, we're going to talk about solving exponential equations with logarithms. So we're going to start by reviewing the relationships for logarithms and then we'll do some examples with a couple of different methods for solving exponential equations with logarithms. 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 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 for my first example, I want to solve the exponential equation, 4 to the x is equal to 50. And I can do this using a logarithm. But before I do that, I want to check to see if I can solve this in my head. In other words, I want to see if 50 is a power of 4.

So is there an integer that I can use as the exponent here? So 4 to some integer, and that will give me a value of 50. So let's think about this.

I know that 4 to the second power is going to be 16. And 4 to the third power is going to give me 64. So I see that my value of x is going to have to be something between 2 and 3 so it's not going to be an integer so that it will evaluate to 50.

So I am going to have to use a logarithm to solve this. So I'm going to start by writing it in logarithmic form. So this will become log base 4 of 50 is equal to x. Now again, I'm going to need a calculator to solve this because I know that my value for x is going to be a decimal.

And on my calculator, I only have usually a logarithmic button for a common log, base 10, or a natural log, which is the base of e. So if I want to use my calculator, I need to convert this logarithmic equation to have either a base 10 or a base of e. And I'm going to use the base of 10, a common logarithm. And I'm going to use my change of base formula to make this into a base 10.

So using my change of base formula, I know that this is going to become log base 10 of 50 over log base 10 of 4. So again, we don't write the base of 10 with a common logarithm. So typing this into my calculator, log base 10 of 50 divided by log base 10 of 4, I see that that gives me a value of 2.82. Approximately 2.82, which is my value for x.

And I can verify that in a couple ways. Number one, this makes sense because if x is 2.82 then I see that that is in between 2 and 3, and it's actually a lot closer to 3, which makes sense because 50 is closer to 64 than to 16. And I can verify that it is even a better estimate by substituting 2.82 in for my variable for x and evaluating that in my calculator.

So I can evaluate 4 to the 2.82 in my calculator. And I see that that does approximately equal 50.

So for my second example, I've got the exponential equation, 7 to the x equals 943. And I want to solve this, again, using a logarithm. So I know that logarithms and exponentials are inverse of each other so to solve this equation, I'm going to use the method of using inverse operations to cancel out the exponent operation.

So I'm going to take the log of both sides of my equation. And this is log base 10. And by doing that, this becomes log of 7 to the x is equal to log of 943. Now I can use the power property of logarithms, which says that this exponent can come to the outside and be a multiplier in front of the logarithm. So now, this becomes x times log of 7 is equal to log of 943.

Now these values, because they're base 10, I can evaluate them in my calculator. So all I need to do is isolate this x variable, and I can do that by dividing both sides by log of 7. So here this will cancel. And now, I'm left with x is equal to log of 943 over log of 7.

And again, because these are both base 10, I can simply type this into my calculator, and I will see that I get a value of approximately 3.52. So I found that 7 to the exponent of 3.52 is approximately equal to 943. And I could verify that in my calculator by just typing in 7 to the 3.52 and see that it does equal approximately 943.

So let's go over our key points from today. Use properties of logarithms to solve exponential equations. We have the power property, the change of base formula, and the conversion between logarithmic form and exponential form.

One method of solving exponential equations involves converting the equation from exponential to logarithmic form, and then using the change of base formula to solve. A second method of solving exponential equations involve taking the log of both sides of the equation and using the power property of logs to simplify and solve.

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

Formulas to Know

- Change of Base Property of Logs
- Power Property of Logs