Combine the three separate logarithms in the expression into a single logarithm. Write all variables with positive exponents. ( logarithms are in the description).

Question

1/2 ( 13 log (x) + 14 log (y) - 5 log (z) )

Answer 1

1/2(13logx + 14logy - 5logz)

Let’s distribute the 1/2 so we can we look at our options:

13/2 logx + 14/2 logy- 5/2 logz

13/2 logx + 7logy - 5/2logz

In order for logs to combine, they must

1. have the same base
2. have no number on the outside (coefficient of 1)

These logs all have the same base of 10, which is the base when it is not written for you. However, they all have numbers outside that we need to get rid of before we can combine them together. Here is how to move those numbers:

Number on outside = exponent on the inside

13/2logx = log(x^13/2)

we can rewrite this in a better way because an exponent fraction with 2 in the denominator means a square root, so this is really log(sqrt x^13)

Let’s do the same process with the second and third logs

7logy = log(y^7)

5/2logz = log(z^5/2) = log(sqrt z^5)

We are one step away from the answer:

log(sqrt x^13) + log( y^7) - log(sqrt z^5)

Now that we have taken care of our bullet points, we can combine the logs together. The rule for combining logs is:

1. logs adding = insides multiply (it becomes a numerator)
2. logs subtracting = insides divide (it becomes a denominator)

The first and second logs are adding so they will be in the numerator of our combined log, while the third log is subtracting and will appear in the denominator. The final answer is:

log( ( sqrt x^13 * y^7 ) / sqrt z^5 )