Evaluating Logarithms

The value of a logarithm can be calculated by two methods:
  • Converting to Exponential Form
  • Using Logarithm Table

Converting to Exponential Form

As dicussed earlier in this post, a logarithmic expression is equivalent to an exponential expression since logarithms are the opposite of exponents.

Suppose we have the logarithmic expression
Let the expression be equal to 'x'. (we have to eventually find the value of 'x'),
`log_{10}(100) = x`
Convert to exponential form,
`10^x = 100`
Write 100 as an exponent of 10. We have `100 = 10 * 10 = 10^2`. Thus,
`10^x = 10^2`
Equate the two expressions. Two exponential expressions are equal when two their bases and exponents are both equal. In the above equation, the bases are 10 in both the expressions and the exponents are `x` and 2. Thus `x` should be equal to 2. Thus,
`x = 2`
Now that we have the value of 'x', we will equate the original expression to 'x',
`log_{10}(100) = 2`

Solved Examples

Solved Example 1
Evaluate the following logarithms:
Let the expression be equal to 'x',
`log_{10}(1000) = x`
Convert to an exponential expression,
`10^x = 1000`
Write 1000 as a power of 10. We have `1000 = 10 * 10 * 10 = 10^3`. Thus,
`10^x = 10^3`
Equate the two exponential expressions. Their bases are both equal to 10. Their exponents are `x` and `3` respectively. Thus,
`x = 3`
Thus, we can write the original expression equal to 'x'.
`log_{10}(1000) = 3`


Using Logarithm Table

Using the logarithm table, we can find the value of logarithms with base 10 and natural base only. However, if you know the base changing formula of logarithms, you can convert a logarithm of any base to logarithms of base 10, evaluate them, and then find the value of the original logarithm with different base.

First we will discuss finding the value of a logarithm with base 10 using a logarithmic table. Suppose you have to evaluate the following logarithm

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