Unit 5: Expanding logarithmic functions

This is an example of a more challenging problem, following the expansion of logarithmic functions method.

Step 1:Step 1: log(4x^4*8y^8÷2z^2)

My first step was to create my original function.

Step 2:log4+logx^4

My next step was to start to expand my equation by separating the first two things that are being multiplied together. This is a step that must be done due to the rules of the product property. First, I have to expand the 4x4by finding the log of both parts individually. I do this, by finding the log4and then adding that to the logx4. Once I do that step, my function has started to expand into the equation shown above.

Step 3: log8+logy^8

I then move on to the next step of the expansion process by separating the 8y8. In order for me to separate it, I have to find the two different parts. Once, I found that they were 8 and y8I found the individual log of each. First I found the log8 and then I added that to the logy^8. After doing that, I was left with the equation shown above.

Step 4:log4+logx^4+log8+logy^8

After completing those two steps I had to combine their two results into the equation above. Due to the fact that they were both in the numerator, I was able to simply use the rules of the product property and connect all four logs by addition.

Step 5:Step 5: log2+logz^2

Next, I had to break down the denominator by separating the different parts. I was able to do this by identifying that it was the product property and then adding the log of the first part to the log of the second part. I found the log2 and then the logz2. Once I did that, I connected them by addition and I was left with the equation above.

Step 6: log4+logx^4+log8+logy^8-log2-logz^2

After completing the last step of separating terms connected by multiplication, I was able to connect the numbers in the numerator and in the denominator. According to the rules of the quotient property, in order to connect the numerator and denominator you must subtract the logs of the denominator from the logs of the numerator. In this case, this meant that I had to subtract the log2+logz^2 from the log4+logx^4+log8+logy^8. This was represented by the equation above.

Step 7: log4+4logx+log8+8logy-log2-2logz

My final step was to use the rules of the power property in order to get rid of the exponents. According to the power property, the only way to get rid of exponents is by moving them to the front of the log. With this prior knowledge, I was able to find each variable that was connected to a variable and I was able to move each exponent to the front.Once I moved all of the exponents one-by-one, I ended up with the equation above.

Step 8:0.60+4logx+0.90+8logy-0.30+2logz

I then had to start to simplify the logs with two numbers, I did so by plugging them in to the calculator. I broke the log4 down to 0.60, the log8 down to 0.90, and the1og2 down to 0.30. After doing that I made the proper adjustments to the equation and I was left with what is shown above.

Step 9: 1.5+4logx+8logy-0.30+2logz

I then had to look for the like terms in order to combine them. I saw that 1.5 and 0.30 were the only individual numbers so I combined them by subtracting them. When I subtracted them I got 1.2.

Step 10: 1.2+4logx+8logy+2logz

Lastly, I replaced the 1.2 in for the two individual numbers. Once I did that, I got the equation shown above.