SOLUTION: write as a single logarithm: 4log(5x+3)-2log(x+15)-1/2log(5x+2)

Question 404103: write as a single logarithm: 4log(5x+3)-2log(x+15)-1/2log(5x+2)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
4log(5x+3)-2log(x+15)-1/2log(5x+2)
These are not like logarithmic terms. Like logarithmic terms have logarithms of the same base and same argument. Your logartihms are all base 10 logarithms but the arguments are all different. Since only like logarithmic terms can be subtracted, we cannot subtract these terms.

There is another way. There are two properties of logarithms:

These properties only require that the logarithms have the same base and coefficients of 1! Your are of the same base but the coefficients are not 1's. Fortunately there is a third property of logarithms, , that allows us to move a coefficient of a logarithm into the argument as its exponent. So we'll use this third property to move the coefficients so that we an then use the other properties to combine them into a single logarithm:

Since an exponent of 1/2 means square root, we can rewrite the last logarithm with a square root:

Now we can use the first two properties. The difference in those two is that the first one has a "+" between the two logarithms and the second property has a "-". Since there is a "-" between the first two logarithms in your expression we will use the second property to combine them:

The remaining logarithms have a "-" between them so we will use the second property again to combine them:

This may be an acceptable answer because it is, after all, a single logarithm. But the argument is a bit of a mess. One does not usually leave expressions with either fractions within fractions or with square roots in denominators. The following steps will "clean up" the argument into a better form:
Multiplying by the reciprocal is the same as dividing:

Now we have a single fraction with no square roots in the denominator. This is a better form than the earlier expression. I doubt that your teacher wants you to multiply out the numerator and denominator.

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