Logarithm

Logarithmic function is the inverse to the exponential function. A logarithm to the base b is the power to which b must be raised to produce a given number. For example, is equal to the power to which 2 must be raised to in order to produce 8. Clearly, 2^3 = 8 so = 3. In general, for b > 0 and b not equal to 1. Fact about Logarithm : 

1. Logarithms were quickly adopted by scientists because of various useful properties that simplified long, tedious calculations.
2. Logarithm to base 10 (that is b = 10) is called the common logarithm and has many applications in science and engineering.
3. Natural logarithm, is a logarithm with base e. It is used in mathematics and physics, because of its simpler derivative.
4. Binary logarithm is a logarithm with base 2 and is commonly used in computer science.

Laws of Logarithms :

Laws	Description
	+
	–
*

How to find logarithm of number? Naive solution: The idea is to create a function that calculates and returns . For example, if n = 64, then your function should return 6, and if n = 129, then your function should return 7. 

C++

// C++ program to find log(n) using Recursion
#include <iostream>
using namespace std;

unsigned int Log2n(unsigned int n)
{
    return (n > 1) ? 1 + Log2n(n / 2) : 0;
}

int main()
{
    unsigned int n = 32;
    cout << Log2n(n);
    return 0;
}

// This code is contributed by Susobhan Akhuli

C

// C program to find log(n) using Recursion
#include <stdio.h>

unsigned int Log2n(unsigned int n)
{
    return (n > 1) ? 1 + Log2n(n / 2) : 0;
}

int main()
{
    unsigned int n = 32;
    printf("%u", Log2n(n));
    getchar();
    return 0;
}

Java

// Java program to find log(n)
// using Recursion
class Gfg1 {

    static int Log2n(int n)
    {
        return (n > 1) ? 1 + Log2n(n / 2) : 0;
    }

    // Driver Code
    public static void main(String args[])
    {
        int n = 32;
        System.out.println(Log2n(n));
    }
}

Python3

# Python 3 program to
# find log(n) using Recursion

def Log2n(n):

    return 1 + Log2n(n / 2) if (n > 1) else 0

# Driver code
n = 32
print(Log2n(n))

C#

// C# program to find log(n)
// using Recursion
using System;

class GFG {

    static int Log2n(int n)
    {
        return (n > 1) ? 1 + Log2n(n / 2) : 0;
    }

    // Driver Code
    public static void Main()
    {
        int n = 32;

        Console.Write(Log2n(n));
    }
}

PHP

<?php
// PHP program
// to find log(n) using Recursion

function Log2n($n)
{
    return ($n > 1) ? 1 + Log2n($n / 2) : 0;
}

// Drive main

    $n = 32;
    echo Log2n($n);
?>

Javascript

<script>

// Javascript program to find log(n)
// using Recursion
function Log2n(n)
{
    return (n > 1) ? 1 + Log2n(n / 2) : 0;
}

// Driver Code
var n = 32;
document.write(Log2n(n));

// This code is contributed by 29AjayKumar

</script>

Output :

5

Time complexity: O(log n) 
Auxiliary space: O(log n) if the stack size is considered during recursion otherwise O(1) 

Efficient solutions:

• Using inbuilt log Function

Practice problems on Logarithm:

Question 1: Find the value of x in equation given 8^x+1 – 8^x-1 = 63
Solution: Take 8^x-1 common from the eq.
It reduce to
8^x-1(8² – 1) = 63
8^x-1 = 1
Hence, x – 1 = 0
x = 1

Question 2: Find the value of x for the eq. given log_0.25x = 16
Solution: log_0.25x = 16
It can be write as
x = (0.25)¹⁶
x = (1/4)¹⁶
x = 4^-16

Question 3: Solve the equation log₁₂1728 x log₉6561
Solution: It can be written as
log₁₂(12³) x log₉(9⁴)
= 3log₁₂12 x 4log₉9
= 3 x 4 = 12

Question 4: Solve for x
log_x3 + log_x9 + log_x27 + log_x81 = 10

Solution: It can be write as
log_x(3 x 9 x 27 x 81) = 10
log_x(3¹ x 3² x 3³ x 3⁴) = 10
log_x(3¹⁰) = 10
10 log_x3 = 10
then, x = 3

Question 5: If log(a + 3 ) + log(a – 3) = 1 ,then a=?
Solution: log₁₀((a + 3)(a – 3))=1
log₁₀(a² – 9) = 1
(a² – 9) = 10
a² = 19
a = √19

Question 6: Solve 1/log_ab(abcd) + 1/log_bc(abcd) + 1/log_cd(abcd) + 1/log_da(abcd)
Solution:
=log_abcd(ab) + log_abcd(bc) + log_abcd(cd) + log_abcd(da)
=log_abcd(ab * bc * cd * da)
=log_abcd(abcd)²
=2 log_abcd(abcd)
=2

Question 7: If xyz = 10 , then solve log(xⁿ yⁿ / zⁿ) + log(yⁿ zⁿ / xⁿ) + log(zⁿ xⁿ / yⁿ)
Solution:
log(xⁿ yⁿ / zⁿ * yⁿ zⁿ / xⁿ * zⁿ xⁿ / yⁿ)
= log xⁿ yⁿ zⁿ
= log(xyz)ⁿ
= log₁₀ 10ⁿ
= n

Question 8: Find (121/10)^x = 3
Solution: Apply logarithm on both sides
log_(121/10)(121/10)^x = log_(121/10)3
x = (log 3) / (log 121 – log 10)
x = (log 3) / (2 log 11 – 1)

Question 9: Solve log(2x² + 17)= log (x – 3)²
Solution:
log(2x² + 17)= log (x² – 6x + 9)
2x² + 17 = x² – 6x + 9
x² + 6x + 8 = 0
x² + 4x + 2x + 8 = 0
x(x + 4) + 2(x + 4) = 0
(x + 4)(x + 2)=0
x= -4,-2

Question 10: log₂(33 – 3^x)= 10^{log(5 – x)}. Solve for x.
Solution: Put x = 0
log₂(33 – 1)= 10^log(5)
log₂32 = 5
5 log₂ 2 = 5
5 = 5
LHS = RHS   

More problems related to Logarithm : 

• Find minimum number of Log value needed to calculate Log upto N
• Find maximum among x^(y^2) or y^(x^2) where x and y are given
• Print all substring of a number without any conversion
• Program to compare m^n and n^m
• Find larger of x^y and y^x

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