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Solving Logarithmic Equations using Exponents

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Hi, and welcome. My name is Anthony Varela and today we're going to solve logarithmic equations. So we're going to review the relationship between exponents and logs, we will rewrite logarithmic equations as exponential equations in order to solve logarithmic equations. And we'll also see how we can raise a base to a logarithmic power in order to solve log equations.

So first, let's review our relationship between exponential equations and logarithmic equations. So here we have y equals b to the x. And I can rewrite this equivalently as log base b of y equals x. So notice that the base of the exponent here is the base to the log, the output of my exponential equation is the input to the logarithmic equation, and my variable power x in my exponential equation is the output to the logarithmic equation.

And I'd also like to review the power rule, because this is very commonly applied to solve a logarithmic equations. And our power rule says that log base b of x raised to the power of n can be rewritten as n times log base b of x. So notice that that exponent n inside of the log function was brought outside, and it is just multiplied by log base b of x.

I'd also like to review change of base. This is particularly useful when using your calculator to evaluate log expressions. And our change of base formula says that log base b of x can be expressed as the log of x divided by the log of b.

So this would be common log of x over common log of b. You can use natural log of x over natural log of b, that's OK too just as long as these two bases are the same. But the change of base is really useful because your calculator probably has the Log button, which is common law base 10 and natural log, which is the log with a base of E. So we're going to write these down as important rules and properties of logs that will help us solve our logarithmic equations.

So let's solve this logarithmic equation, log base 7 of x equals 3. Now to solve this one, what I'm going to do is just rewrite this as an exponential equation. So I'm going to take the base of my log, that's 7, that's going to become the base of my exponential equation.

Remember that the output of the log equation is the exponent to my base. So I have 7 raised to the power of three, and that equals x. So solving this is actually pretty simple, I am just going to evaluate 7 cubed. Well, that's 343. So log base 7 of 343 equals 3.

So let's solve a more complicated logarithmic equation. Here we have log base 5 of 2x minus 3 equals 1.2. So the strategy that I'm going to employ here is I'm going to take the base to my log, 5, and I'm going to raise this to a power. And that power is going to be this side of the equation.

So I have 5 raised to log base 5 of 2x minus 3. Now to do the same thing to my other side in order to keep things equal, I'm going to have my base of 5 raised to the power of 1.2, taking this entire expression here.

So now what I've done looking at a base 5 being raised to the log base 5 of something, logs and exponents are inverse operations. So this actually cancels out. And what I'm left with on this side of the equation then is just what's inside of the log. So that would be 2x minus 3.

So interestingly enough then, this side of the equation simplifies to 2x minus 3, what was ever inside of my log equation, which was an exponent of base 5. And this only works then when the base year in the base year match up. Well, now I can evaluate 5 to the power of 1.2, that's 6.9 when I round.

And now this is an equation that I can solve pretty easily. It's a linear equation. So I'm going to add 3 to both sides, 2 x equals 9.9, and then divide by 2. So I've solved for x as being equal to 4.95. Now we can go ahead and confirm this solution.

So what I'm saying then is that log base 5 of 2x minus 3 equals 1.2 when x equals 4.95. So plugging in 4.95 for x, we're saying that the log base 5 of 6.9 is 1.2. I'm going to use the change of base to write this as an expression I can type in into my calculator.

So to evaluate log base 5 of 6.9, I'm going to say that this is log 6.9 divided by log 5. And that has to equal 1.2. So typing log 1.69 into my calculator dividing that by a log of 5, I have confirmed that 1.2 is 1.2.

So let's review our lesson on solving logarithmic equations. Well, there's this relationship between exponents and logs that might help you solve some logarithmic equations. If we have y equals b to the x, we can write this as log base b of y equals x.

We often employ the power rule. So this allows us to take an exponent inside of a log function and bring it outside and multiply it by log base b of x. And we often use the change of base formula, which says that log base b of x equals log x over log b. So thanks for watching this tutorial on solving logarithmic equations, hope to see you next time.