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Logarithm

  • Difficulty Level : Medium
  • Last Updated : 30 Jun, 2021

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, \log_2 8  is equal to the power to which 2 must be raised to in order to produce 8. Clearly, 2^3 = 8 so \log_2 8  = 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 : 
 

LawsDescription
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\log_a b  \log_a c
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\log_a b  – \log_a c
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c\log_a b
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-\log_a b
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0
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1
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r
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*



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\log_a c
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1/\log_a b

How to find logarithm of number?
Naive solution: 
The idea is to create a function that calculates and returns \log_2 n  . 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 <bits/stdc++.h>
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)<<"\n";
    return 0;
 
// This code is contributed by UJJWAL BHARDWAJ

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);
?>

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: 
 

 

Practice problems on Logarithm:

Question 1: Find the value of x in equation given 8x+1 – 8x-1 = 63 
Solution: Take 8x-1 common from the eq. 
It reduce to 
8x-1(82 – 1) = 63 
8x-1 = 1 
Hence, x – 1 = 0 
x = 1 
Question 2: Find the value of x for the eq. given log0.25x = 16 
Solution: log0.25x = 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 4log9
= 3 x 4 = 12 
Question 4: Solve for x 
logx3 + logx9 + logx27 + logx81 = 10
Solution: It can be write as 
logx(3 x 9 x 27 x 81) = 10 
logx(31 x 32 x 33 x 34) = 10 
logx(310) = 10 
10 logx3 = 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/logab(abcd) + 1/logbc(abcd) + 1/logcd(abcd) + 1/logda(abcd) 
Solution: 
=logabcd(ab) + logabcd(bc) + logabcd(cd) + logabcd(da) 
=logabcd(ab * bc * cd * da) 
=logabcd(abcd)2 
=2 logabcd(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)
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 – 3x)= 10log(5 – x). Solve for x. 
Solution: Put x = 0 
log2(33 – 1)= 10log(5) 
log232 = 5 
5 log2 2 = 5 
5 = 5 
LHS = RHS
  
More problems related to Logarithm : 
 

Recent Articles on Logarithm!
 

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