Logarithms: which exponent is needed

Going from 1 to 1000 takes three multiplications by 10. A logarithm asks which exponent produces a number, including fractional and negative exponents. Use inverse functions and exponentials.

$\log_a b=c\ \Longleftrightarrow\ a^c=b;\qquad a>0,\ a\ne1,\ b>0.$

Examples: $\log_{10}1000=3$ , $\log_{10}0.1=-1$ , $\log_2\sqrt2=1/2$ . Real logarithms exclude zero and negative arguments. The graph of logₐx reflects aˣ across y=x: domain (0,∞), range ℝ; increasing for a>1, decreasing for 0<a<1. Its zero is x=1; x=0 is a vertical asymptote.

Why a product becomes a sum

Write x=aᵘ and y=aᵛ. Then xy=aᵘ⁺ᵛ, so for x,y>0:

$\log_a(xy)=\log_a x+\log_a y,\qquad \log_a(x/y)=\log_a x-\log_a y.$

Likewise a power multiplies the exponent. Taking logarithms of $a^{\log_a x}=x$ changes the base:

$\log_a(x^r)=r\log_a x\ (x>0),\qquad\log_a x=\frac{\ln x}{\ln a}.$

Here ln has base e. No base turns a logarithm of a sum into a sum of logarithms: $\log_{10}(10+10)\approx1.301$ , but $\log_{10}10+\log_{10}10=2$ . OpenStax

Linear scale

Logarithmic scale

Dimensionless values. Top: distance proportional to v; bottom: to log₁₀v. Equal lower steps multiply by 10 rather than add a constant. Zero is excluded from the logarithmic axis.

Equal distances mean equal factors

On a base-10 logarithmic axis, 0.1,1,10,100 are equally spaced: their coordinates are −1,0,1,2. Halfway between 1 and 100 lies 10, not 50.5. For physical quantities, logarithms require a dimensionless ratio to a stated unit or reference. Our numbers are dimensionless; decibels or earthquakes would require a separate domain model.