3.2 Logarithmic and exponential functions

For positive arguments, log(ab)=log a+log b, log(a/b)=log a−log b and log(a^p)=p log a. Change of base converts any legal base to a calculator base.

Record argument restrictions before combining or expanding. A logarithm base must be positive and not equal to 1.

ln[(x−1)^2] can be written 2ln|x−1| for x≠1, not 2ln(x−1) without the domain restriction.

Logarithms do not distribute over addition, and dropping absolute values can exclude valid negative arguments.

e^{ln x}=x for x>0 and ln(e^x)=x for real x. Exponential and logarithmic graphs reflect in y=x, with domains and ranges exchanged.

When solving, isolate the exponential or logarithm, then apply the inverse and preserve positivity. For models, identify whether the exponent is linear in the independent variable.

e^{2x−1}=7 gives x=(1+ln7)/2; a logarithm of a non-positive quantity is not defined over the reals.

ln(e^{x}) and e^{ln x} have different domain conditions even though both simplify algebraically.

Solve logarithmic equations by combining legal logs, converting to an exponential equation and checking every solution in the original domain.

If a quadratic appears after exponentiating, both algebraic roots are only candidates; reject any that make a log argument zero or negative.

log₃(x)+log₃(x−2)=1 requires x>2. It becomes x(x−2)=3, whose roots are 3 and −1; only x=3 survives.

Exponentiating an equation preserves equivalence only when both sides were defined; it cannot legalise an invalid log argument.

Rewrite a nonlinear relation as Y=mX+c, then plot the transformed variables. The gradient and intercept must be converted back to the original parameters.

Choose transformations from the model, not from the data alone; label axes and preserve units. A straight plot tests compatibility with the model over the measured range.

For y=a/x+b, plotting y against 1/x gives gradient a and intercept b.

A high correlation after transformation does not prove the mechanism, and the fitted intercept may be a transformed constant.