Condensing
1/2logs+logm+logh-2logI-logd-logk
logt+logh+logx-2logl-log0-logl
To condense the expressions given, we need to use several properties of logarithms.
For the first expression:
1/2log(s) + log(m) + log(h) - 2log(I) - log(d) - log(k)
We can use the following properties:
1. Product Rule: log(a) + log(b) = log(ab)
2. Power Rule: log(a^b) = blog(a)
3. Quotient Rule: log(a/b) = log(a) - log(b)
By applying these properties, we can condense the expression:
1/2log(s) + log(m) + log(h) - 2log(I) - log(d) - log(k)
= log(sqrt(s)) + log(m) + log(h) - log(I^2) - log(d) - log(k)
= log(sqrt(s)) + log(m) + log(h) - 2log(I) - log(d) - log(k)
= log(sqrt(s) * m * h / (I^2 * d * k))
For the second expression:
log(t) + log(h) + log(x) - 2log(l) - log(0) - log(l)
Here, we encounter an issue because the logarithm of zero is undefined. Therefore, we need to revise the expression:
log(t) + log(h) + log(x) - 2log(l) - log(0) - log(l)
= log(t) + log(h) + log(x) - 2log(l) - log(l)
= log(t * h * x / (l^2))
Note that we cannot combine the terms with "log(l)" into a single term because the coefficient in front of log(l) differs.
In summary:
1/2log(s) + log(m) + log(h) - 2log(I) - log(d) - log(k) = log(sqrt(s) * m * h / (I^2 * d * k))
log(t) + log(h) + log(x) - 2log(l) - log(0) - log(l) = log(t * h * x / (l^2))