Logarithm Question for Pre-Calculus Algebra?
I have the basics down, but I have 2 more questions on my homework. I can't figure out these. If someone answers these can they please show the steps - I need to be able to do these on my own on the exam.
Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)
Log Base 8 x^5 Sqrt y/z^5
Condense the expression to the logarithm of a single quantity.
1/2[logbase6 (x+5) + 2 logbase6(x-3)] + 3 logbase6 x
I sat with a tutor on Friday at my school, but I guess she didn't know what she was doing either because the answers we got together were wrong.
Thanks Again!
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Verified answer
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1. log₈(x⁵(√y)/z⁵) ← Given the problem instructions and no parentheses
to show grouping, I'm guessing that this is the problem.
= log₈(x⁵) + log₈(√y) - log₈(z⁵)
= log₈(x⁵) - log₈(z⁵) + log₈(y^(½))
= 5 log₈(x) - 5 log₈(z) + ½ log₈(y) ← This could be the answer, but, there are
other variations that could be in your
answer key. Here are a few of those:
➊ 5 [log₈(x) - log₈(z)] + ½ log₈(y)
➋ 5 log₈(x/z) + ½ log₈(y)
➌ ½ [10 log₈(x/z) + log₈(y)]
Note: The answers you got with your tutor
may well be correct, but, need to be
manipulated to match the answer key.
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2. ½ [log₆(x+5) + 2 log₆(x-3)] + 3 log₆x ← First, distribute the ½
= ½ log₆(x+5) + log₆(x-3) + 3 log₆x
= log₆[(x+5)^½ ] + log₆(x-3) + 3 log₆(x³)
= log₆√(x+5) + log₆(x-3) + log₆(x³)
= log₆[(x-3)√(x+5) / x³] ← ANSWER
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