Math Help!?

Question

Find X:
log3(-9)=x <---that is log base 3. I can't do a little 3.

2^x=3

2.43 times 10^x = 1.84

Also, What is the range of the function f(x)= -|x+3|-2

Thank you soo much!!

Answer 1

(1) log3(-9) = x

Are you allowed to use complex numbers in this answer? Without them, you can't do a log of a negative number.

x = log3(-9)
x = log3(9 * (-1))
x = log3(9) + log3(-1)
x = 2 + ln(-1)/ln(3)
x = 2 + i*pi/1.0986
x = 2 + 2.8596 * i

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(2) 2^x = 3
ln(2^x) = ln(3)
x ln(2) = ln(3)
x = ln(3) / ln(2)
x = 1.5849825

Checking answer:
2^1.5849825 = 3.00004
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(3) 2.43 times 10^x = 1.84
10^x = 1.84 / 2.43
ln(10^x) = ln(1.84 / 2.43)
x ln(10) = ln(1.84) - ln(2.43)
x = (ln(1.84) - ln(2.43)) / ln(10)
x = -0.12078845

Checking answer:
10^-0.12078845 * 2.43 = 1.840000002

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(4) The absolute value is greater than or equal to zero, always. Since that absolute value is subtracted from -2, the range is f(x) <= -2

f(x) = -2 when x = -3, and decreases linearly (with slope 1 and -1) on both sides of that point.