# Logarithm Formulas

## Formulas for Logarithms

When the power of a number must be raised in order to get some other number known as Logarithm. Here on this page, you will get Logarithm Formulas which will help you to Solve Logarithms based Questions easily.

### Formulas for Logarithm:-

• Definition & Logarithm Formulas:

Logarithms are the power to which a number is raised to achieve some other number.

• Logarithms is of 2 types:-
• Common logarithm
• Natural logarithm.
• Common Logarithm-

Logarithm with base 10 is Common logarithm.

It is expressed as log10 X,  and if any expression is not given with the base, then the base 10 is considered.

• Natural Logarithm-

Logarithm with base e is Natural Logarithm.

It is expressed as logX.

• Very Important: If the base is not provided, then always remember to consider base as 10.

### Formulas for Logarithm

• logx X= 1
• loga 1= 0
• a loglogax =X
• logax =  $\frac{1}{log_x a} \$
• loga(x p) = p(log ax)
• loga x = $\frac{log_{10} X}{log_{10} a} \$
• loga$\frac{x}{y} \$= loga X- loga Y
• loga (xy)= logaX+ logaY

### Value of log(2 to 10): Remember

• Log 2 = 0.301
• Log 3 = 0.477= 0.48
• Log 4 = 0.60
• Log 5 = 0.698 = 0.7
• Log 6 = 0.778 = 0.78
• Log 7 = 0.845 = 0.85
• Log 8 = 0.90
• Log 9 = 0.954= 0.96
• Log 10 = 1

### Logarithm Formulas (Antilog):

• An antilog is the inverse function of a logarithm.
log(b) x = y means that antilog (b) y = x.
• The best way to understand any problem is by having a look at the Solved Example.
• We are going to do the same here, and we are going to understand the Antilog problem by Solved Example.

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### Sample Logarithm Questions with Solution

Question: 1 Calculate the antilog of 3.6552.

Solution:

As we know that antilog is the inverse of the logarithm. so, it is clear that we are going to find the number whose logarithm is 3.6552

From the antilog table,  we would find the value corresponding to row 65 and column 5 is 4508.

The mean difference column for the value 2 is 2.

Adding these two values, we have 4518 + 2 = 4520.

The decimal point is placed in 3 + 1 = 4 digits from the left. So, antilog 3.6552 = 4520.0

Question: 2 What will the value of x in the equation $2^x=16$ ?

Solution:

In logarithm notation: $\log_{2}16=x$

$2^4=16$
$\log_{2}(2^4)$

x=4

Question: 3 Evaluate the value of x in the equation: $\log_{5}x+\log_{5}2=3$

Solution:

Using the logarithm property,

$\log_{a}x+\log_{a}b=\log_{a}(x.b)$
$\log_{5}(x.2)=3$
$x.2=5^3$
$x.2=125$
$x=\frac{125}{2}=62.5$

Question: 4 Evaluate the value of x in the equation $\log_{3}x+\log_{3}2=1$

Solution:

Using the logarithm property,
$\log_{a}x-\log_{a}b=\log_{a}(\frac{x}{b})$
$\log_{3}(\frac{x}{2}=1$

$\frac{x}{2}=3^1$
$\frac{x}{2}=3$

To solve for x, multiply both sides by 2:
x=6

Question: 5 Solve for x: ln(2x + 1) = 3

Solution:

Convert the equation to exponential form according to the natural logarithm,

$e^3 = 2x + 1$

$\frac{(e^3-1)}{2}$

So, x ≈ 3.1945

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