Page 368 - Instant notes

Appendix 354

Sums, products and differences

∆x=(final value of x)−(initial value of x)

Logarithms and exponentials

x
If y=a , then log a(y)=x
a is the base of the log. Logs in base e are denoted as ln(y).
Mathematical definitions require that x and y must both be dimensionless.
In the absence of a subscript, log(x) usually implies log 10(x) and ln(x) denotes log e(x)
Basic relationships:
log(xy)=log(x)+log(y)
log(x/y)=log (x)−log(y)
y
log(x )=y log(x)
Converting from base a to base b:
log a(x)=log a(b) . log b(x)
Hence, log e (x)=2.303 log 10(x)

Differentials
In all cases, a and c are independent of x, i.e. a≠f(x) and c≠f(x).
Basic differential relationships:

Differentials of products, sums, and quotients (f, g, u, υ are all functions of x):