If log(a) 2 + log(a) 4 = log (2) a^2, what is a?
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If log(a) 2 + log(a) 4 = log (2) a^2, what is a?
We have log(a) 2 + log(a) 4 = log (2) a^2
convert all the terms to a form with logarithm to the base 2
=> log 2 / log a + log 4 / log a = log a^2 / log 2
=> (1/ log a)( log 2 + log 4) = 2*log a / log 2
Use log a + log b = log a*b and log 2 = 1
=> log 8 / log a = 2* log a
=> log 8 = 2* (log a)^2
=> 2* (log a)^2 = 3* log 2 = 3
=> (log a)^2 = (3/2)
=> log a = sqrt (3/2)
a = 2^(sqrt (3/2))
a = 2^(sqrt 1.5)
The required value of a is 2^(sqrt 1.5)
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calendarEducator since 2008
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log(a) 2 + log(a) 4 = log(2) a^2 .
We will use logarithm properties to find a.
We know that log a + log b = log a*b
==> log(a) 2 + log(a) 4 = log(a) 2*4 = log(a) 8
==> log(a) 8 = log(2) a^2
Now we will rewrite:
log(a) 8 = log(2) 8 / log(2) a = log(2) 2^3 / log(2) a = 3/log(2) a
==> 3/log(2) a = log(2) a^2
==> 3 = 2log(a) 2 * log(2) a
==> 2[log(2) a)'^2 = 3
==> log(2) a = sqrt(3/2)= sqrt(1.5)
==> Now we will rewrite into exponent form.
==> a = 2^(sqrt(1.5)
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