Question 160200: 3log x = log125
Here is another one I can not find a practice problem like it.
Thank you in advance for your help!
Found 2 solutions by Alan3354, gonzo:
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! 3log x = log125
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3log(x) = log(x^3) (that's how logarithms work, you multiply the log by the exponent)
So, log(x^3) = log(125)
Then x^3 = 125
x = 5 (the principal root, there are 2 others that I don't recall how to find. email me if you need them.)
Answer by gonzo(654)  (Show Source):
You can put this solution on YOUR website! here's what i think.
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log 125 without telling you what the base is implies the base is 10.
making that assumption, i went to the calculator and found the log of 125.
it is 2.096910013
if log (125) = y, and log(125) = 2.096910013, then y = 2.096910013.
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i looked at the other equation of 3*log(x) and again assumed base of 10.
the basic properties of logarithms state that 3*log(x) is the same as log (x^3).
your equation then becomes log(x^3) = log(125) = 2.096910013 = y.
the basic properties of logarithms also state that y = log(x) if and only if 10^y = x.
this means that y = log(x^3) if and only if 10^y = x^3.
since y = 2.096910013, then the equation becomes
10^(2.096910013) = x^3.
since 10^(2.096910013) = 125, the equation becomes
125 = x^3
since cube root of (125) is 5, then
x = 5
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substituting for x in the original equation of 3*log(x) = log(125), we get
3*log(5) = log(125)
log(5) = .698970004
log(125) = 2.096910013
substituting for log(5) and log(125), the equation becomes
3*.698970004 = 2.096910013
this becomes 2.096910013 = 2.096910013
answer checks out
3*log(5) = log(125)
x = 5
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keys to solving this were:
1. know that log without specifying the base assumes a base of 10.
2. know that 3*log(x) = log(x^3)
3. know that y = log(x) if and only if 10^y = x
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another way of looking at this that might have been even simpler.
it's pretty logical actually.
i'm not sure if it works in all cases but it worked in this one.
it goes as follows:
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if log(a) = log(b) then a must = b
this probably would work with any base, not just 10, as long as the base on the left side of the equation is the same as the base on the right side of the equation.
correspondingly, if log(a^3) = log (b) then a^3 must = b
that would have led directly to x^3 = 125 from the original equation.
you would have had to know, at least, that 3*log(x) = log(x^3).
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both ways lead to the same answer, at least in this case.
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