# Logarithms

## Definition

The logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the solution y to the equation.

If any base b is equal to the anti-logarithm, the logarithm is always 1.

If the anti-logarithm is 1, the logarithm is always 0.

## Formula

1. Product: The logarithm of a product is the sum of the logarithms of the numbers being multiplied.
2. Quotient: The logarithm of the ratio of two numbers is the difference of the logarithms.
3. Power: The logarithm of the p-th power of a number is p times the logarithm of the number itself.
4. Root: The logarithm of a p-th root is the logarithm of the number divided by p.

### Change of base

The logarithm can be computed from the logarithms of x and b with respect to an arbitrary base k.

Given a number x and its logarithm logb(x) to an unknown base b, the base is given by:

## Exercises

### Q.1

Solve the exponential equation. $2^x = 3^{x-1}$:

### Q.2

Plot running time T(N) vs. input size N using log-log scale. $\log_{2}(T(N)) = b\log_{2}N + c$: