To determine the value of [tex]\( a \times b \times c \)[/tex], we need to solve each of the given equations and then find the product of the solutions.
1. Solving for [tex]\( a \)[/tex] in the equation [tex]\( 20^a = 16 \)[/tex]:
We can solve for [tex]\( a \)[/tex] using logarithms. Take the logarithm of both sides of the equation:
[tex]\[
\log(20^a) = \log(16)
\][/tex]
Use the logarithmic identity [tex]\( \log(x^y) = y \log(x) \)[/tex]:
[tex]\[
a \log(20) = \log(16)
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{\log(16)}{\log(20)}
\][/tex]
Knowing that [tex]\( 16 = 2^4 \)[/tex] and using the property [tex]\( \log(2^4) = 4 \log(2) \)[/tex]:
[tex]\[
a = \frac{4 \log(2)}{\log(20)}
\][/tex]
2. Solving for [tex]\( b \)[/tex] in the equation [tex]\( 10^b = 20 \)[/tex]:
Similarly, take the logarithm of both sides:
[tex]\[
\log(10^b) = \log(20)
\][/tex]
Apply the logarithmic identity:
[tex]\[
b \log(10) = \log(20)
\][/tex]
Since [tex]\( \log(10) \)[/tex] is 1 (because [tex]\( \log_{10}(10) = 1 \)[/tex]):
[tex]\[
b = \frac{\log(20)}{\log(10)}
\][/tex]
Therefore:
[tex]\[
b = \log_{10}(20)
\][/tex]
3. Solving for [tex]\( c \)[/tex] in the equation [tex]\( 4^c = 10 \)[/tex]:
Take the logarithm of both sides:
[tex]\[
\log(4^c) = \log(10)
\][/tex]
Apply the logarithmic identity:
[tex]\[
c \log(4) = \log(10)
\][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[
c = \frac{\log(10)}{\log(4)}
\][/tex]
Knowing that [tex]\( 4 = 2^2 \)[/tex] and using the property [tex]\( \log(2^2) = 2 \log(2) \)[/tex]:
[tex]\[
c = \frac{\log(10)}{2 \log(2)}
\][/tex]
4. Finding the product [tex]\( a \times b \times c \)[/tex]:
Now we multiply the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
a \times b \times c = \left(\frac{4 \log(2)}{\log(20)}\right) \times \left(\frac{\log(20)}{\log(10)}\right) \times \left(\frac{\log(10)}{2 \log(2)}\right)
\][/tex]
Simplify the expression by canceling out the common logarithmic terms:
[tex]\[
a \times b \times c = \left(\frac{4 \log(2)}{\log(20)} \cdot \frac{\log(20)}{\log(10)} \cdot \frac{\log(10)}{2 \log(2)}\right)
\][/tex]
Notice that [tex]\( \log(20) \)[/tex] and [tex]\( \log(10) \)[/tex] cancel out:
[tex]\[
a \times b \times c = \left(\frac{4 \log(2)}{\log(20)} \cdot \frac{\log(20)}{\log(10)} \cdot \frac{\log(10)}{2 \log(2)}\right) = \frac{4}{2} = 2
\][/tex]
So, the value of [tex]\( a \times b \times c \)[/tex] is:
[tex]\[
\boxed{2}
\][/tex]
