In Exercises 41–70, use properties of logarithms to condense each...

Question

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator.

2 logb x + 3 logb y

Accepted Answer

Hello. Today we're going to be combining the given logarithms into a single logarithms. So the given logarithms is four times the logarithm of base X. Of A plus seven times the lagoon rhythm of base X of C. Now we don't want to have the coefficients of four and seven. So we need to get we need to find a way to get rid of these. Now we can do that using the exponential rule for logarithms. And that states that if you have the logo rhythm of A. To the power of B. This can be rewritten as B. Times the logo rhythm of A. So using this property going from right to left, we can go ahead and take the coefficients of four and seven and make them exponents of the inside arguments of the logarithms. So once we apply the exponential rule to both our logarithms. What we are left with is the logarithms of base X. Of A. To the power of four plus the logarithms of base X of C. To the power of seven. Okay. And now we need to go ahead and combine both of these logarithms together notice that we have an addition sign here. So this is an indication that we should use the multiplication multiplication property for logarithms. And that states that if you have the logo rhythm of A times B, then this can be rewritten as the logo rhythm of A. Plus the logo rhythm of B. And once again we're gonna be using this property going from right to left. Here are a is going to be A. To the power of four, and R. B. Is going to be C. To the power of seven. And applying the property here is going to give us the longer rhythm of base X. Of A. To the power of four times C. To the power of seven. And this is going to be the complete simplification of the given logarithms. And that means the answer to this problem is going to be C. So I hope this video helps you in understanding how to combine logarithms together. And I'll go ahead and see you all in the next video.

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In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 2 logb x + 3 logb y