What is the logarithm of log1/2 4=x?

log₁/₂ 4 = x
½^x = 4
(2^-1)^x = 2^2
2^(-x) = 2^2
-x = 2
x = -2

log₈ 4 = x
8^x = 4
(2^3)^x = 2^2
2^(3x) = 2^2
3x = 2
x = 2/3

log₁₀ 0.001 = x
10^x = 0.001
10^x = 10^(-3)
x = -3

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Alternative methods:

Change of base

x = log₁/₂ 4
x = log₂ 4 / log₂ ½
x = log₂ 2² / log₂ 2⁻¹
x = 2/-1
x = -2

x = log₈ 4
x = log₂ 4 / log₂ 8
x = log₂ 2² / log₂ 2³
x = 2/3

is this saying log base 1/2 of 32=x ? because if it is a logarithm is and exponent to which you need to raise the base to get the number so if 1/2 is the base and x is the exponent of one half then its 1/2 to the x power = 32 right? so yeah it's -5 1/2 to the -5 power is 32, so x = -5 does that help any?

1)
log_(1/2)(4) = x
(1/2)^x = 4
(2^-1)^x = 2^2
2^(-x) = 2
-x = 2
x = 2/(-1)
x = -2

2)
log_8(4) = x
8^x = 4
(2^3)^x = 2^2
2^(3x) = 2
3x = 2
x = 2/3

3)
log_10(0.001) = x
10^x = 0.001
10^x = 1/1000
10^x = 1/(10^3)
10^x = 10^(-3)
x = -3

________________________
(1)
log½ 4 = x

... (½)^x = 4 ... ← by definition of log
.. (2ֿ¹)^x = 2²
... 2^(-x) = 2² ← this is true only if the exponents are equal
... .... -x = 2
.... .... x = -2


________________________
(2)

log8 4 = x

.. 8^x = 4 ... ← by definition of log
(2³)^x = 2²
2^(3x) = 2² ... ← this is true only if the exponents are equal
..... 3x = 2
.... .. x = 2/3

_________________________
(3)
log10 0.001=x

10^x = 0.001 ... ← by definition of log
10^x = 1/1000
10^x = 1/10³
10^x = 10ֿ³ ... ← this is true only if the exponents are equal
.... x = -3


_______________________________

THE LOG IS THE EXPONENT

log8 4 = x means 8^x = 4, so you see 8 as 2³ and 4 as 2² and so
[2^(3)]^x = 2²
2^(3x) = 2^2
3x = 2
x = 2/3
think it through a few times and you'll be able to solve these at a glance
log_27 81 = 4/3, for instance, since 81 = 3^4 and 27 = 3^3

log10 0.001 = log10 10^-3 = -3, easy, trivial

log_1/2 4 = X
the same as 1/2^(x) = 4
you could rewrite it as 2^ (-x) = 4

2 to the second is obviously 4, so -x = 2, x= -2

as for the others

log_8 4 = x
simply use a calculator using conversion formula:
log 4 / log 8

app. 2/3

log_10 .001 = x
log_10 = the common log = log
log .001 = x
log (1000^-1) = x
-1 log (1000) = x
-1 (3 ) = x
x = -3

log1/2 4=x => (1/2)^x = 4 = 2^2 = (1/2)^-2 => x = -2

log1/2 4=x
is basically asking 1/2 to the x power equals 4 which is -2

log8 4=x
8 to the x power equals 4 which is, well i got around 2/3

log10 0.001=x
for this one, since log has the base of 10 you can just plug this into the calculator. depends on which type of calculator u have. for TI 30 you need to put in .001 then push the log button. for TI 84 push the log button then put in .001.

Question 1
(1/2) ^x = 4
2^(-x) = 2^2
x = - 2

Question 2
8^x = 4
x log 8 = log 4
x = log 4 / log 8
x = 2 / 3

Question 3
10^x = 0.001
10^x = 10^(-3)
x = - 3