Find the numerical value of the log expression.
log a=8 log b=-6 log c=6
log b^3c^7/sqrt a^3
Using the properties of logarithms, we can rewrite the expression as:
log b^3c^7/sqrt a^3 = (log b^3 + log c^7 - log (sqrt a^3))
Substituting the given values:
= (3(log b) + 7(log c) - log (sqrt a^3))
= 3(-6) + 7(6) - log (sqrt 8^3)
= -18 + 42 - log (2^3)
= -18 + 42 - log (8)
= -18 + 42 - 3
= 21 - 3
= 18
Therefore, the numerical value of the log expression is 18.
To find the numerical value of the log expression, we need to use the properties of logarithms. Specifically, we will use the power rule and the quotient rule.
Let's break down the expression step by step:
1. Start with the expression: log(b^3c^7/sqrt(a^3))
2. Apply the power rule: log(b^3) + log(c^7) - log(sqrt(a^3))
3. Rewrite the square root using the exponent: log(b^3) + log(c^7) - log(a^3)^(1/2)
4. Use the power rule to rewrite the exponent: log(b^3) + log(c^7) - log(a^(3/2))
5. Apply the quotient rule: [log(b^3) + log(c^7)] - log(a^(3/2))
6. Use the properties of logarithms to simplify further: [3 * log(b) + 7 * log(c)] - (3/2) * log(a)
Now, plug in the values for log(a), log(b), and log(c) given in the problem:
[3 * (-6) + 7 * 6] - (3/2) * 8
Multiply and simplify:
[-18 + 42] - (3/2) * 8
[24] - (3/2) * 8
24 - 12
12
Therefore, the numerical value of the log expression log(b^3c^7/sqrt(a^3)) is 12.
To find the numerical value of the log expression, we can substitute the given values of log a, log b, and log c into the expression.
The given values are:
log a = 8,
log b = -6,
log c = 6.
Substituting these values into the expression log b^3c^7/sqrt a^3, we get:
log b^3c^7/sqrt a^3 = log b^3 + log c^7 - log (sqrt a^3)
Now, let's substitute the given values:
= log (b*b*b) + log (c*c*c*c*c*c*c) - log (a^(3/2))
= 3 log b + 7 log c - log (a^(3/2))
= 3(-6) + 7(6) - log (a^(3/2))
= -18 + 42 - log (a^(3/2))
= 24 - log (a^(3/2))
Since log a = 8, we can substitute this value:
= 24 - log (a^(3/2))
= 24 - log (sqrt a^3)
= 24 - log (sqrt (a*a*a))
= 24 - log (a^(3/2))
= 24 - log (a^1.5)
= 24 - 1.5 log a
= 24 - 1.5 * 8
= 24 - 12
= 12
Therefore, the numerical value of the given log expression, log b^3c^7/sqrt a^3, is 12.