Logarithms: Properties, Equations, and Change of Base

A logarithm answers an exponent question. The statement ${\log }_{b}x=y$ means that raising the base $b$ to exponent $y$ produces $x$. For real logarithms, the base must be positive and different from one, and the argument must be positive. These restrictions are part of the definition, not optional checks added after a calculation.

Base, argument, and logarithm each have a different role

In ${\log }_{2}8=3$, 2 is the base, 8 is the argument and 3 is the logarithm. The equality is true because $2^3=8$. A logarithm may be negative: ${\log }_{10}0.01=-2$. It cannot have zero or a negative real argument. Base ten is often called the common logarithm, while base $e$ is the natural logarithm and is written $\ln$.

The notation $\log$ without a displayed base depends on context. Many school texts mean base ten; higher mathematics and software may mean the natural logarithm. State the convention or use ${\log }_{10}$ and $\ln$ explicitly when ambiguity matters.

Compare an exponential and its logarithm

Change the base. The two curves reflect across the line y = x because the functions undo one another.

BaseSelected x-value

Product, quotient, and power rules come from exponent laws

For positive arguments, the logarithm of a product is a sum, the logarithm of a quotient is a difference, and an exponent can move in front as a multiplier. There is no corresponding rule that splits the logarithm of a sum. In general, ${\log }_b(x+y)$ is not ${\log }_{b}x+{\log }_{b}y$.

These identities condense products or expand them for equation solving. Apply the domain first. For example, the expression $\log (x-2)$ requires $x>2$; algebra performed outside that domain does not create a valid logarithm.

Change of base lets a calculator handle any valid base

Divide logarithms in the same convenient base: ${\log }_{b}x=\frac{\ln x}{\ln b}$. The denominator is nonzero because $b\ne 1$. A calculator with only $\ln$ and ${\log }_{10}$ can therefore evaluate ${\log }_{2}7$. The choice of auxiliary base cancels as long as the same one is used in numerator and denominator.

Worked example: solve a sum of two logarithms

Solve ${\log }_2(x-1)+{\log }_2(x-3)=3$. The arguments first require $x>3$. Combine the sum into the logarithm of a product, convert to exponential form, and solve the resulting quadratic. The algebra gives two candidates, but only one lies in the original domain.

The candidates are 5 and negative 1. Substitution rejects negative 1 because both logarithmic arguments would be negative. The only real solution is $x=5$, and checking gives ${\log }_{2}4+{\log }_{2}2=2+1=3$.

Logarithms reverse compound growth

When the unknown appears in an exponent, a logarithm can isolate it. In the compound-interest equation $M=C{(1+i)}^n$, the period count is $n=\frac{\log \left(\frac{M}{C}\right)}{\log (1+i)}$, provided all required quantities are positive and $i>-1$, with $i\ne 0$ so the denominator is nonzero. This connects logarithms to compound growth without turning the mathematical result into a forecast.

On a graph, an exponential function and its same-base logarithm are reflections across $y=x$. For bases greater than one both increase; for bases between zero and one both decrease. The logarithm has a vertical asymptote at $x=0$, which marks the excluded boundary of its domain.