simplify log 10^36 - 2(10^log 12)
a. -108
b. 0.25
c. 1.5
d. 12
log 10^36 - 2(10^log 12)
10^llogX = X so 10^log12 = 12
so
36 log 10 - 24
but log 10 is1
36 -24 = ?
by the definition of log,
10^logx = x
log 10^x = x
So, your expression is just
36-2(12)
To simplify the expression log(10^36) - 2(10^log12), we can use the properties of logarithms.
First, let's simplify log(10^36). According to the basic property of logarithms, log(a^b) = b * log(a). Therefore, log(10^36) = 36 * log(10).
The logarithm of 10 to any base is equal to 1. So, log(10) = 1.
Substituting this value back into our expression, we have log(10^36) - 2(10^log12) = 36 * 1 - 2(10^log12).
Next, let's simplify 10^log12. According to the inverse property of logarithms, log(a^b) = b * log(a). Therefore, log(10^log12) = log12 * log(10).
Again, log(10) = 1, so log(10^log12) = log12 * 1 = log12.
Substituting this value back into our expression, we have 36 - 2log12.
Finally, we just need to evaluate 2log12. Since we already know the logarithm of 12, we can simply multiply it by 2.
Therefore, 2log12 = 2 * log12 = 2log(10^log12) = 2log12 = 24.
Substituting this value back into our expression, we have 36 - 24 = 12.
So, the simplified expression is 12.
Therefore, the answer is d) 12.
To simplify the given expression,
log 10^36 - 2(10^log 12)
We can start by using the rule that says log base b of a raised to the power of n is equal to n times log base b of a. Therefore, we can rewrite the expression as:
36log 10 - 2(log 12)
The log base 10 of 10 is equal to 1, so we can simplify further:
36 - 2(log 12)
To simplify the expression within the parentheses, we can use logarithmic properties.
Remember that log base b of c multiplied by d is equal to log base b of c plus log base b of d.
Therefore, we can rewrite the expression as:
36 - 2(log 3 + log 4)
Now, we can use another logarithmic property which states that log base b of c plus log base b of d is equal to log base b of c times d.
Therefore, we have:
36 - 2(log 3 × 4)
Simplifying further:
36 - 2(log 12)
Now, we can substitute log 12 with its logarithmic value.
The logarithmic value of 12 can be obtained by using the change of base formula. We can choose any base to convert to, let's choose base 10:
log 12 = log base 10 of 12 / log base 10 of 2
The logarithmic value of 12 is approximately 1.0792.
Substituting back into our expression:
36 - 2(1.0792)
Now, we can simplify the multiplication:
36 - 2.1584
Finally, subtracting:
36 - 2.1584 ≈ 33.8416
Therefore, the simplified value is approximately 33.8416.
None of the options provided match this value, so it seems that there may be an error in the given options. Please double-check the options or the original expression.