Hi, this is Anthony Varela. And today we're going to solve exponential equations using logarithms. So we'll review the relationship between exponents and logs. Then we're going to solve exponential equations using two methods.
The first, we're going to rewrite the equation as a logarithmic equation. In our second example, we're going to take the log of both sides of our equation, because exponents and logarithms are inverse operations.
So let's review this relationship between exponents and logs. Here we have an exponential equation, y equals b raised to the power of x. So the base number b being raised to a variable power.
Well, we can rewrite this equation equivalently using a logarithm. We can say log base b of y equals x. So let's talk about where these pieces moved as we go from an exponential equation to a logarithmic equation.
Well, the output of our exponential y, that is the input of the logarithmic equation. Our base number here is the base of the log. And then this variable power is the output of our logarithmic equation. So that's the relationship between exponents and logs.
And we're going to be using two important properties of logs to help us solve our exponential equations. The first is called the power rule. And so if we have the log of base b of an expression that has an exponent included within the argument of this log function, what we could do is bring that exponent out and multiply it by log base b of x. So the exponent can be brought out and is no longer an exponent, but it's a multiplier of the log expression.
We're also going to be using change of base, which means if we have an expression, log base b of x, we can rewrite this as log of x divided by log of b. And we use this because our calculators operate in common log-- that's log with base 10. So this is the common log of x divided by the common log of b. And so we can type in just log x in our calculator and get a value, divide that by log b into our calculator to solve our equations.
So we're going to be using these two properties as we solve our exponential equations. So our first example, we'd like to solve 6 raised to the power of x equals 9.39.
Now I'm going to have to use logarithms. You might be aware of the strategy of trying to rewrite this number to have a base of 6, and then just set the exponents equal to each other. But 9.39 is not a power of 6. So we're going to be using logarithms to solve for x.
And so what I'm going to do is rewrite this as a log equation using a relationship that we talked about just moments ago. So I'm going to write this then as the log base 6 of 9.39 equals x. Remember, we're taking the log of this output. It has a base of 6. And the variable exponent is the output of our log equation.
So now how can we evaluate log base 6 of 9.39? Well, we're going to be using then our change of base so that we can type some values into our calculator. So this is going to be the common log of 9.39 divided by the common log of 6. That's applying our change of base.
So plugging that into our calculator then, log of 9.39 over log 6, that equals x. And we get a value of 1.25. So you can check that by raising 6 to the power of 1.25. And you'll get 9.39.
Our next example, we have a base of 5.5 raised to the x power. And this equals 21.5. So what we're going to do now is take advantage of the fact that exponents and logarithms are inverse operations.
And so we're going to take the log of both sides of the equation. Very similar to if you wanted to undo something being multiplied by 10, you would divide by 10. So we want to undo this exponent. So we're going to apply the logarithm.
So we're going to say that the log of 5.5 raised to the x power equals the log of 21.5. So we're taking this entire expression, putting it inside of a log function, taking this entire expression, putting it inside of the logarithm.
So on the left side of our equation, then we see we can apply the power rule. We have an exponent inside of a logarithm. So we're going to move that out. So we have x multiplied by the log of 5.5 equals the log of 21.5.
Remember, these are just numbers. So we can divide both sides by log 5.5. And it looks very similar to what we did before with our change of base where we have the log of this expression divided by the log of this base number. So looks very much the same.
Typing this into our calculator, we see that x equals 1.8. We can check this by raising 5.5 to the power of 1.8. And we'll get 21.5.
So let's review our lesson on solving exponential equations using logs. Well, we started with this relationship between logs and exponents. So an exponential equation, y equals b to the x, can be rewritten as a log equation, log base b of y equals x.
We used the power rule in our first example. So if you spot an exponent inside of a log expression, that can be pulled outside of a log expression. And it acts then just as a multiplier.
And we also used the change of base formula. And this really helped us be able to use our calculator to evaluate logarithms. Our calculator's log button assumes base 10. So if we have base b, we can take the common log-- that's base 10 of x-- over the common log base b.
So thanks for watching this tutorial on solving exponential equations using logarithms. Hope to see you next time.