Hello!
Could someone please take a look at the problem below and let me know if I made mistakes in simplifying the given equation? I'm quite certain that I did make a mistake somewhere because the original equation and its simplified form give different graphs when plotted. I would greatly appreciate any help!
y=1/4log2(8x-56)^16 -12
y=(16)(1/4)log2(8x-56)-12
y=4log2(8x-56)-12
y=4log2(8x) : 4log2(56) -12
y=log2(8x)^4 : 4log2(56) -12
y=log2((8^4)(x^4)) : 4log2(56) -12
y=log2(8^4) + log2(x^4) : 4log2(56) -12
y=log2(2^12) + log2(x^4) : 4log2(56) -12
y=12 + log2(x^4) : 4log2(56) -12
y=log2(x^4) : 4log2(56)
y=4log2(x) : 4log2(56)
y=4log2(x-56)
Your major mistake is in line 4
You are basically saying
log(A-B) = logA/logB
there is no such relation!
From line 3:
y=4log2(8x-56)-12
= 4log2(8(x-7)) - 12
= 4log28 + 4log2(x-7) - 12
= 4(3) + 4log2(x-7) - 12
= 4log2(x-7) , where x>7
Yes you are right, there is no such relation. Thank you so much for your help!
Let's go through the steps and check for mistakes:
1. Original equation: y = 1/4log2(8x-56)^16 - 12
2. Simplification: y = (16)(1/4)log2(8x-56) - 12 (Correct)
3. Simplification: y = 4log2(8x-56) - 12 (Correct)
4. Simplification: y = 4log2(8x) : 4log2(56) - 12 (Mistake)
The mistake appears here. When dividing the expression, the denominator should be included in parentheses as it affects both terms. The correct step should be y = 4log2(8x) : (4log2(56) - 12).
5. Simplification: y = log2(8x)^4 : (4log2(56) - 12) (Correct)
6. Simplification: y = log2((8^4)(x^4)) : (4log2(56) - 12) (Correct)
7. Simplification: y = log2(8^4) + log2(x^4) : (4log2(56) - 12) (Correct)
8. Simplification: y = log2(2^12) + log2(x^4) : (4log2(56) - 12) (Correct)
9. Simplification: y = 12 + log2(x^4) : (4log2(56) - 12) (Correct)
10. Simplification: y = log2(x^4) : (4log2(56)) (Mistake)
Another mistake occurs here. The denominator should be 4log2(56) - 12 and not just 4log2(56).
11. Simplification: y = 4log2(x) : (4log2(56) - 12) (Correct)
12. Simplification: y = 4log2(x-56) (Correct)
So, the mistake occurred in steps 4 and 10. The correct simplified equation is y = 4log2(8x) : (4log2(56) - 12) and not y = 4log2(x-56).
To verify if you made mistakes in simplifying the given equation, we can step through your simplifications and compare them to the original equation. Let's analyze each step:
Original Equation: y = (1/4)log2((8x-56)^16) - 12
Step 1: y = (16)(1/4)log2(8x-56) - 12
You correctly simplified the exponent by using the property that (a^b)^c = a^(b*c). The "16" and "1/4" cancel out, yielding 4log2(8x-56) - 12.
Step 2: y = 4log2(8x-56) - 12
This step is correct. No errors here.
Step 3: y = 4log2(8x) : 4log2(56) - 12
Here, it seems like there might be a mistake. It's unclear what you intended by adding ": 4log2(56) - 12" at the end of the equation. If you meant to write it as a fraction, it should be written with a horizontal line instead of a colon. But to match the original equation, we can omit this part.
Step 4: y = 4log2(8x) - 12
This is the same as step 2, so it's correct.
Step 5: y = log2(8x)^4 : 4log2(56) - 12
Here, it looks like you tried to apply the power rule of logarithms, which states that log(a^b) = b * log(a). However, you applied it incorrectly. The correct simplification should be: y = 4log2(8x^4) : 4log2(56) - 12.
Step 6: y = 4log2(8x^4) : 4log2(56) - 12
This step is correct. No errors here.
Step 7: y = log2((8^4)(x^4)) : 4log2(56) - 12
Here, you correctly simplified the exponent, resulting in y = log2(4096x^4) : 4log2(56) - 12.
Step 8: y = log2(8^4) + log2(x^4) : 4log2(56) - 12
This step is correct. No errors here.
Step 9: y = log2(2^12) + log2(x^4) : 4log2(56) - 12
Here, you simplified 8^4 to 2^12, which is correct. No errors here.
Step 10: y = 12 + log2(x^4) : 4log2(56) - 12
This step is correct. No errors here.
Step 11: y = log2(x^4) : 4log2(56)
Here, you omitted the "+12" term from the original equation. Including it would yield:
y = log2(x^4) + 12 : 4log2(56)
Step 12: y = 4log2(x) : 4log2(56)
Here, you mistakenly wrote "log2(x^4)" as "4log2(x)". The correct simplification should be:
y = log2(x^4) : 4log2(56)
Looking closely at Step 12, it seems like there was an error in this step. To accurately simplify the equation, you can't cancel out the logarithms in this way.
Therefore, there is a mistake in Step 12, and the correct simplification is:
y = log2(4096x^4) : 4log2(56) - 12
I hope this clarifies the errors you made in simplifying the equation.