 Some mathematicians are predictable enough that it should come as no surprise that after introducing a new thing, the logarithm, we should be interested in trying to solve equations involving logarithms, and so we can take a look at logarithmic equations. And so a logarithmic equation is an equation that includes a logarithmic expression of our variable. So a general way to solve these is we can solve these by using the product quotient and power rules for logs, specifically log of a product equals the sum of logs, log of the quotient is the difference of logs, log of the power, and so on and so forth. We can use our rules of logarithms to transform our equation into some form log equals something, log to base a of b equals n. And at that point, again, definitions are absolutely important. You cannot do mathematics without knowing the definitions. The definition of log to base a of n means that I can rewrite this as an exponential expression and eliminate the logarithmic expression. So again, the base is the base, the argument is the power, and the log itself is the exponent. And that will allow me to eliminate the logarithm. For example, let's take log to base 2, 2x plus 10 equals 3 plus log to base 2 of x minus 4. And again, a little analysis goes a long way. The right hand side is a sum, so we can subtract the add-end. So I'll subtract log to base 2 of x minus 4, and the left hand side is the difference of two logs. Log something minus log something. My quotient rule says anytime I have a difference of logs, I can write it as a quotient of two logarithms. So that's going to be log of the quotient 2x plus 10 over x minus 4. And now I have this in the form log equals number. And so my definition of logs, log to base a of b is equal to n, tells me that a, log to the base to the log is equal to the argument. So I have 2 to the power 3 is equal to 2x plus 10 over x minus 4. And I have a rational equation which I can now solve. And so solving my rational equation, and I get a solution x equals 7. Now because this is a logarithmic equation, it's, well actually I do need to check my solution in the original equation because I want to make sure I haven't introduced an extraneous solution through our equation solving process. So I'll check to make sure that x equals 7 does in fact solve the original equation. So I'll substitute into my original equation. So that's log to base 2, 2x plus 7, 3 plus log to base 2 of x minus 4. And so there's my first expression, parentheses say do stuff inside first. So I take care of 7 minus 4, I take care of 2 times 7 plus 10. And let's see. I can evaluate this, I can at least find the difference of logs. So let's see, that's a difference of two logs, that's a quotient, 24 over 3, that's a equal to 8, log to base 2 of 8, does it actually equal 3? And so the thing we might remember 2 to power 3 is equal to 8. And so this statement, the log is the exponent, this is a true statement. So x equals 7 is our solution to this equation.