Converting Logarithms to Exponential Form

Consider the following logarithm equation:

`log_{10}(100) = 2`

and the following exponential equation:

`10^2 =100`

We can say that both the above equations are equivalent. They are just different forms of the same equation.

Every logarithmic expression can be converted to an exponential expression by remembering the following rules:

• The base of the logarithm is the base of the exponential expression
• The number/quantity which the logarithmic expression equals is the exponent in the exponential expression
• The argument of the logarithm is the result of the exponential expression 

Thus, relating to the above two equations, we can write the following steps:

1. Base of the logarithm `log_{10}(100)` is 10, hence 10 will be the base of the exponential expression.
2. The logarithmic expression `log_{10}(100)` equals 2. Thus, 2 is the exponent. That is, `10^2`
3. The argument of the logarithmic expression is 100. This becomes the result of the exponential expression. Thus, we can write `10^2 = 100`.

Following the above three steps, you can convert any logarithmic expression/equation to its equivalent exponential expression/equation. A more detailed post on this topic is available here.

Solved Example

Convert the following logarithmic equation to exponential form:

`log_{2}(32) = 5`

First, identify the base, argument and result of the logarithmic equation:

• Base = 2
• Argument = 32
• Result = 5

Now using the rules described above, we know the following things about the exponential expression:

• The base 2 of the logarithm will become the base of the exponential expression
• The argument 32 of the logarithm will become the result of the exponential expression
• The result 5 of the logarithmic equation will become the exponent

Thus, the exponential equation equivalent to the above equation is,

`2^5 = 32` 

Worksheet

Instructions

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Question

Answer

`log_2(3) = 8`

`2^3 = 8`

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`log_{10}(1000) = 3`

`10^3 = 1000`

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`log_{10}(10) = 1`

`10^1 = 10`

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`log_{5}(25) = 2`

`5^2 = 25`

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`ln(e) = 1`

`e^1 = e` (hint: base of natural logarithm is always the natural number `e`)

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