2.2 Logarithmic and exponential functions

For a,b>0 and base k>0,k≠1, log_k(ab)=log_k a+log_k b, log_k(a/b)=log_k a−log_k b and log_k(a^p)=p log_k a.

Check every argument is positive before combining logs. Change of base gives log_k a=ln a/ln k and is useful when the calculator uses natural logs.

log₂(8x)−log₂x=3 for x>0, because the ratio is 8.

log(a+b) is not log a+log b; the product law does not apply to sums.

For a>0,a≠1, y=a^x has inverse x=log_a y. The natural logarithm ln x is log_e x, and ln(e^x)=x for real x while e^{ln x}=x only for x>0.

Use the inverse relation to solve for a variable, and preserve positivity when taking logarithms. Exponential growth has constant proportional rate; logarithmic growth has decreasing gradient.

3e^{2x}=12 gives e^{2x}=4 and x=½ln4.

ln(x²)=2lnx only when x>0; for x≠0 the safe statement is ln(x²)=2ln|x|.

To solve a logarithmic equation, combine logs only when arguments are positive, then exponentiate or change base. Every candidate must satisfy the original log domains.

If log terms have different bases, convert consistently. Squaring or exponentiating can create algebraic candidates that the original equation rejects.

ln(x−1)+ln(x+1)=ln3 becomes ln(x²−1)=ln3 with x>1, so x=2; x=−2 is rejected by the domain.

Equality of log expressions does not allow non-positive arguments, even if a later algebraic step produces a real number.

Transform variables so the model becomes Y=mX+c. For y=ab^x, plotting ln y against x gives gradient ln b and intercept ln a; for y=ax^n, plotting ln y against ln x gives gradient n.

Transform measured uncertainties and units consistently, then interpret the gradient and intercept in the original parameters. A straight plot supports the model but does not prove causation.

If ln y=0.7x+1.2, then a=e^{1.2} and b=e^{0.7} for y=ab^x.

The intercept is not always the original constant; it may be ln a or another transformed quantity.