Respuesta :
Answer: Neither equation is true
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Explanation:
We can pick various values for m and x to show that log(mx) = m*log(x) is not true overall.
For instance, consider m = 2 and x = 10
- log(mx) = log(2*10) = log(20) = 1.301 approximately
- m*log(x) = 2*log(10) = 2*1 = 2 exactly
The equation log(mx) = m*log(x) becomes 2 = 1.301 after plugging in m = 2 and x = 10. We get different values on either side, so that shows the original equation is false for those particular values. More broadly, the original equation is false for all real numbers m and x.
In short, log(mx) is not the same as m*log(x).
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The same idea can be done for the second equation.
Let's say we go for n = 3 and x = 100
- log(x^n) = log(100^3) = 6
- ( log(x) )^n = ( log(100) )^3 = 2^3 = 8
The second equation boils down to 6 = 8 when we plug in the mentioned n and x values. This is one counterexample showing that the second equation is false overall. It may work for a special set of n and x values, but it doesn't work for every real value.
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Side notes:
- I'm using log base 10 for each section above. The same idea works for other bases as well.
- One handy log rule is log(AB) = log(A)+log(B)
- Another log rule is log(A/B) = log(A)-log(B)
- Yet another rule is log(A^B) = B*log(A)
