Logarithms are an integral part of the calculus. To solve a logarithm without a calculator, let us first understand what a logarithm is.
Defining a logarithm or log
A logarithm is defined as the power or exponent to which a number must be raised to derive a certain number. The number that needs to be raised is called the base.
Let’s see how a logarithm is depicted: c
Here “x” is the base.
Now, let’s get to the main part:
How to Solve a Log Without Using a Calculator?
We first need to understand square, cubes, and roots of a number. This is key to solving a logarithm.
The solution of any logarithm is the power or exponent to which the base must be raised to reach the number mentioned in the parenthesis.
log x (y) = z
If xz = y, then ‘z’ is the answer to the log of y with base x, i.e., log x (y) = z
In other words, x needs to be raised to the power z to produce y. z is hence the answer to log x (y).
Here log x (y) is known as the logarithmic form, and xz = y is known as the exponential form. Remembering and understanding this equivalency is the key to solving logarithmic problems.
log x (y) = log x (xz) = z
Let us use an example to understand this further: log 5 (25)
The base in this logarithm is 3. Let us try to replace the number in the parenthesis with the base raised to an exponent.
log 5 (25) = log 5 (52)
One the base and the number in the parenthesis are identical, the exponent of the number is the solution to the logarithm.
Therefore log 5 (25) = 2.
Some more examples:
log 2 (32) = log 2 (25) = 5
log 6 (1) = log 6 (60) = 0
log 4 (16) = log 4 (42) = 2
It is to be noted that in some instances you might notice that the base is not mentioned. Example: log 1000. In such cases, it is understood that the base value by default is 10.
So log 1000 = log 10 (1000) = 3.
Some logarithms are more complicated but can still be solved without a calculator.
Here are some examples:
log 4 (1/64)
log 1/4 (64)
log 121 (11)
log 3/2 (27/8)
Let us solve each one of these.
Let us consider that log 4 (1/64) equals to z
This can be written in another form as: 4z = 1/64
Now let us try to find z, by simplifying the equation
4z = (1/43)
4z = 4-3
Hence z = -3
log 1/4 (64) = z
Here 64 needs to be converted to (1/4) raised to an exponent, which is the solution to the logarithm.
(1/4)z = 64
(1/4)z = 43
4-z = 43
Hence z = -3
log 121 (11) = z
To find z, first let us convert this to exponential form: 121z = 11
We know that 121 is 11 squared, and hence the square root of 121 is 11.
√121 = 11
1211/2 = 11
Hence z = ½
log 3/2 (27/8) = z
This equation is not as difficult as it may seem.
Let us convert it to exponential form (3/2)z = (27/8)
(3/2)z = (3/2)3
Hence z = 3
log 2√32 = z
This can be rewritten as log 2 (32)1/2 = z
In the exponential form, this is equivalent to 2z = 321/2
We know that 25 = 32
2z = (25)1/2
2z = (25/2)
Hence z = 5/2
Solving a logarithm without a calculation is easier than it might seem. The most crucial part is to be well versed with squares, cubes, and roots of numbers.
The other important part of solving a logarithm is understanding its exponential form. Once you can do this, with a little practice, you can easily solve logarithms without needing a calculator.
Do you have any logarithms you are unable to solve? Or do you have any questions for us? Please mention in the comments section below, and we will be happy to assist you.