SL 1.7—Rational exponents and logarithm laws

Apply rational exponent and logarithm laws consistently.

Fractional powers represent roots and powers, while log laws convert products, quotients and powers into sums, differences and coefficients.

Worked example
x^(3/2)=(√x)³ for x≥0; log(ab)=log a+log b for positive a,b.

Worked example
When can you split log(ab)? both arguments must be in the valid domain.

Common boundary
Do not apply log laws to sums or ignore domain restrictions.

Use all three log laws only for positive arguments: $\log_a(xy)=\log_ax+\log_ay$, $\log_a(x/y)=\log_ax-\log_ay$, and $\log_a(x^m)=m\log_ax$. Change base with $\log_ax=\ln x/\ln a$. Example: $2^{x-1}=10$ gives $(x-1)\ln2=\ln10$, so $x=1+\ln10/\ln2\approx4.322$. Never split $\log(x+y)$.