Value of Log 2

The formal definition of logarithms is that it is the inverse of exponentiation. This is very important to often make the plots of data easier to understand. To assess the quality of sound, the Richter scale to measure the impact of an earthquake, etc we use a logarithmic scale. The two most important logarithms we use are natural logarithms or the logarithms to the base ‘e’ and logarithm to the base 10. Natural logarithms are often written ln(x) whereas log(x) often means logarithm to the base is 10.

Logarithms have proven to be a frequent sticking point for math students over the years. Often, they are part of these students' introduction to the world of exponents. Many of the concepts aren't intuitive and don't necessarily follow from anything else the students may have learned about mathematics.

Value of Log

Let us try to understand how to do logarithmic operations using the example log2. An exponential is written of the form xn = m. This means if we multiply x, n times we get m. For example 23 = 8. 

Now if we take the logarithm of m to the base x we get n. That means n is the exponential of x to obtain m. This is written as follows: Logx(m) = n. So taking the previous example, we get log2(8) = 3. The log value of a number depends on the base.

Log2x in Action

When you explore log problems with bases other than 10, none of the aforementioned principles change. The math can look a little wonkier, so take care not to confuse small bases like 2 with whatever the log is, as these numbers are often in the low single digits, too.

Example: What is log24,000?

The answer completes the sentence "4,000 is the result of 2 being raised to the power of..." The value of this expression is 11.965.

You can use an online tool like the one in the Resources instead of your calculator to solve log2 problems.

Some Important Properties of Logarithm

The following properties of logarithm help us to solve problems easier.

• Logb(b) = 1.

This property is very easy to understand. We know if bn = b, then n should be 1. Hence, the property.

• logb(MN) = logb(M) + logb(N) .

This is called the product rule. Let us try to understand Why this rule works. Suppose bm = M and bn = N. Then, b(m+n) = MN.

Taking logarithm of MN, we get logb(MN) = m+n (from the definition of logarithm). Now logb(M) = m and logb(N) = n.

That is, logb(MN) = logb(M) + logb(N)

• Logb(am) = m*logb(a)

This is known as the law of exponents.

Logb(am) = logb(a*a*a*….(m times)).From the product rule we know this is equal to

Logb(a) + logb(a) +……. m times = m*logb(a)

• Logb(M/N) = logb(M) – logb(N)

• Logb(m) = loga(m)/loga(b). This is known as the change – of –base formula. 

• The logarithm of 0 or negative numbers is not defined. 

• The logarithm of a number between 0 and 1 gives a negative value.

• The logarithm of 1 is 0.

Examples of Logarithmic Problems

Given y = log10100,000, what is y?

y is the exponent to which 10 must be raised to get the value 100,000. This is 5, as you may be able to do in your head if you know that 105 = 100,000.

Given y = log1050,000, what is y?

y is the exponent to which 10 must be raised to get the value of 50,000. Clearly, this is a noninteger value since 104 = 10,000 and 105 = 100,000. Your calculator can provide the answer: 4.698. (This is a good reminder that exponents do not have to be whole numbers.)

Logarithmic Expressions

Say you are asked to solve the expression y = log101000. First, you need to identify what is happening in the problem. When you get a value for y, it has to be an exponent.

To be precise, it is the exponent (or power) to which the base (given as a subscript and taken to be 10 when not explicitly given) must be raised to get the argument of the log, which is the only number you see in the standard form at the start of these problems.

That is, the above expression is equivalent to 10y = 1,000. You may recognize on sight that y must be equal to 3, but if not you can rely on your calculator to get the correct answer.

Value of Log(2)

As stated earlier to find the value of the log of any number we should know the base. Here it is always easier to find the value of log(2) to the base 10 and use it to find the value of the logarithm of 2 on any other basis. Using properties we can find the answer through calculations, close to the actual value.

We know 210 is 1024. Taking the logarithm, we get

10 *log(2) = log(1024).

Approximating 1024 to 1000, 

Log(2) = \[\frac{log(1000)}{10}\] = \[\frac{log(10^{3})}{1}\]. From exponential rule, we can write this as \[\frac{3log(10)}{10}\].

Since the base is also 10, we get log(2) = 3*0.1. = 0.3. This is a very accurate value as the value we obtain using a calculator is 0.301. 

We can use the expansion formula of the natural logarithm to find the value of ln(2).

ln x = Substituting the value 2 in place of x, we obtain the value of ln(x) = 0.69.

Using these values we can find the value of other logarithms too.

We have discussed the basic meaning of logarithms, their important properties, and formulas and how to find values of logs of numbers in brief.

How can I find the value of log 2 without a calculator?

Base 10 or natural log?

Base 10. 210=1024,10log2=3210=1024,10log⁡2=3+log1.024log1.024

Which gives 0.3 as a pretty good approximation. You can use that log 2 = 0.3, to work out that log 1.25 = 1–3log 2 = 0.1. This suggests that log 1.025 = 0.01, and use this circular reasoning to refine your estimate of log 2 to 0.301.

If you have some time on your hands. ln2=−∑(−1)nn

How can I find the value of log 2 without a calculator?

Base 10 or natural log?

Base 10. 210=1024,10log2=3210=1024,10log⁡2=3+log1.024log1.024

Which gives 0.3 as a pretty good approximation. You can use that log 2 = 0.3, to work out that log 1.25 = 1–3log 2 = 0.1. This suggests that log 1.025 = 0.01, and use this circular reasoning to refine your estimate of log 2 to 0.301.

If you have some time on your hands. ln2=−∑(−1)nn

How to Calculate Antilog?

The antilog of any number is just the base raised to that number. So antilog10(3.5) = 10(3.5) = 3,162.3. This applies to any base; for example, antilog73 = 73 = 343.

You can also obtain the value of the antilog of a number from its logarithmic expression. For example, log101,000,000 = 6, making the antilog of 6 to the base 10, which you can also write log10-1(6), equal to 1,000,000, or the argument of the log expression.