There is an exponent in the middle term which can be brought down as a coefficient. We begin by taking the three things that are multiplied together and separating those into individual logarithms that are added together. Since we are trying to break the original expression up into separate pieces, we will be using our properties from left to right. We have now condensed the original problem into a single logarithmic expression. We can combine those into a single log expression by multiplying the two parts together. Now there are two log terms that are added. We usually begin these types of problems by taking any coefficients and writing them as exponents. Since this problem is asking us to combine log expressions into a single expression, we will be using the properties from right to left. Use the properties of logs to write as a single logarithmic expression.But if you are trying to break up a single log expression into its separate parts, you’ll want to use the property from left to right. If you are being asked to combine log expressions into a single expression, you’ll want to use the property from right to left. So if properties 3, 4 and 5 can be used both ways, how do you know what should be done? That depends on the type of problem that is being asked. ![]() You can also go the other way and move a coefficient up so that it becomes an exponent. It allows you to take the exponent in a logarithmic expression and bring it to the front as a coefficient. This property will be very useful in solving equations and application problems. Two log expressions that are subtracted can be combined into a single log expression using division. This property allows you to take a logarithmic expression involving two things that are divided, then you can separate those into two distinct expressions that are subtracted. Two log expressions that are added can be combined into a single log expression using multiplication. This property allows you to take a logarithmic expression of two things that are multiplied, then you can separate those into two distinct expressions that are added together. This property says that if the base and the number you are taking the logarithm of are the same, then your answer will always be 1. You can verify this by changing to an exponential form and getting. This property says that no matter what the base is, if you are taking the logarithm of 1, then the answer will always be 0. You can verify why this works by changing to an exponential form and getting and anything to the zero power is 1. In addition to the property that allows you to go back and forth between logarithms and exponents, there are other properties that allow you work with logarithmic expressions. This problem is nice because you can check it on your calculator to make sure your exponential equation is correct. In the logarithmic form, the 625 will be by itself and the 4 will be attached to the 5. The 625 was attached to the 5 and the 4 was by itself. Since the base is the same whether we are dealing with an exponential or a logarithm, the base for this problem will be 5. So when we switch those values, the 2 is by itself and the 8 is “attached” In this case the 2 is “attached” to the 3 and the 8 is by itself. It doesn’t matter if x and y are variables or numbers. ![]() In the exponential form in this problem, the base is 2, so it will become the base in our logarithmic form.īecause logarithms and exponents are inverses of each other, the x and y values change places. ![]() ![]() When changing between logarithmic and exponential forms, the base is always the same.
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