Converting Logarithmic to Exponential Equation

Logarithms are the opposite of exponents. Thus, a logarithmic equation can be converted to an equivalent exponential equation.

`log_2(8) = 3` is the same equation as `2^3 = 8`.

Any logarithmic equation has three parts:

  1. Base of logarithm (the number written in subscript after logarithm; 2 in the above example)
  2. Argument of logarithm (the number written after the base in line with 'log'; 8 in the above example)
  3. Result (the number after the = sign; 8 in the above equation)
Similarly, any exponential equation has three parts:
  1. Base of exponent (2 in the above example)
  2. Exponent (or power, or index; 3 in the above example)
  3. Result (8 in the above example)
To convert a logarithmic equation to an exponential equation, we use the following concepts:
  1. Base of logarithm becomes base of exponent
  2. Argument of logarithm becomes result of exponential equation
  3. Result of logarithmic equation becomes exponent
Consider the logarithmic equation `log_{10}(100) = 2`,
  1. Base is 10
  2. Argument is 100
  3. Result is 2
According to the concepts explained above, we will change the above numbers as follows:
  1. Base of logarithm, 10, becomes base in the exponential equation
  2. Argument 100 becomes result of exponential equation
  3. Result of logarithmic equation, 2, becomes exponent
Thus, the equivalent exponential equation to `log_{10}(100) = 2` is `10^2 = 100`.

Difference between Sequence and Progression

A sequence is a set of numbers in a specific order such that you can predict the next number by a specific logic. For example, the following set of prime numbers:
2, 3, 5, 7...
 is a sequence, because
  • it is a set of numbers
  • separated by commas
  • in order of a specific logic (numbers are prime)
On the other hand, a progression is a sequence of numbers which has a specific mathematical formula. That is, you can predict the next term of a progression by the help of a formula, but you can't always have a formula to predict the next number of a sequence.

The set of prime numbers is a perfect example to differentiate between sequences and progressions; It is no doubt that the set of prime numbers is a sequence, but it is not a progression because there is no specific formula to predict the next prime number.

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

Click on Show Answer button to show a particular answer. Click on the answer itself to hide it. Use the buttons at the bottom to show/hide all answers
Question
Answer
`log_2(3) = 8`
Show Answer
`log_{10}(1000) = 3`
Show Answer
`log_{10}(10) = 1`
Show Answer
`log_{5}(25) = 2`
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`ln(e) = 1`
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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
`log_{10}(100)`
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:
`log_{10}(1000)`
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`

Worksheet

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
`log_{10}(100)`

Base and Argument of Logarithm

What are Base and Argument in a Logarithm?

Consider the following expression:
`log_{10}(2)`
In the above expression,
  • The number 10 is called the base of the logarithm
  • The number 2 is called the argument of the logarithm
The above expression is read as
Logarithm of 2 to the base 10.
We can see that
  • The base is always written in subscript (below) just after the word 'log'
  • The argument is the number written in brackets after the base. It is written in line with the word 'log'

What do the base and argument mean in a logarithm?

If we refer to a logarithm table, the value of `log_{10}(2)` is 100. That is,
`log_{10}(2) = 100`
We already know that logarithms are the opposite of exponents. The above expression if converted to exponents is
`10^2 = 100`
In the above expression, we know that 10 is the base raised to the power (exponent) of 2. Thus we call 10 the base of the logarithm `log_{10}(2)`

The number 2 is called the argument because logarithm (to the base 10) is a function to which we pass the value '2' in order to get the result 100.

What if you change the base of the above logarithm?

Suppose we change the base of the above logarithm to 5. What will the result be then?
`log_5(2) = ?`
Let the result be 'x'.
`log_5(2) = x`
Since 5 is the base and 2 is the argument, we will raise 5 to the power of 2 to get 'x'.
`5^2 = x`
Evaluate `5^2 = 5 * 5 = 25`, thus,
`5^2 = 25`
Thus x is 25 and the logarithm's result is 25.
`log_5(2) = 25`
Thus, base of a logarithm is as important as its argument. Changing it changes the meaning and value of the logarithm.

Special Cases

  1. If the base of a logarithm is not specified, we assume it to be 10. Thus, `log(2)` is same as `log_{10}(2)`
  2. If the base of a logarithm is not specified and instead of 'log' there is 'ln' (which means natural logarithm) then the base is the exponential constant, `e`. Thus, `ln(10)` is same as `log_e(10)`.

Solved Examples

Identify the base and argument of the following logarithms:
  1. `log_{10}(5)`
  2. `log_{100}(2)`
  3. `log_x(y)`
  4. `log(10)`
  5. `ln(100)`

Answers:

  1. The number written in subscript after log is the base. Thus, 10 is the base. The argument is the number written in line with log after the base. Thus, 5 is the argument.
  2. 100 is the base and 2 is the argument.
  3. The variable 'x' is written in subscript after log. Thus the base is `x`. `y` is written after the base and it is in line with 'log'
  4. No base is specified. Thus the base is 10.
  5. No base is specified and `ln` is written instead of `log`. Thus, the base is the exponential constant, `e`.

Worksheet

Instructions: 

Find the base of the following logarithms

Click on Show Answer button to show a particular answer. Click on the answer itself to hide it. Use the buttons at the bottom to show/hide all answers
Question
Answer
`log(3)`
Show Answer
`log_{100}(50)`
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`log_e(10)`
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`log_{a}b`
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`log_{2^3}(8)`
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Instructions:

Find the argument of the following logarithms:

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Question
Answer
`log(e)`
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`log(10)`
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`log_{10}(5)`
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`log_{n}(2)`
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`ln(2)`
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General Form of a Quadratic Expression

A quadratic expression written in the following format is said to be in the general form.
`ax^2 + bx + c`

Characteristics of the general form:

  • `a`, `b` and `c` are numbers
  • `a` does not equal 0
  • `b` and `c` can be equal to 0

Description of the general form:

When a quadratic equation or any algebraic equation is written in descending order of exponents of its variable, it is said to be in the general form.

The above equation can be written as
`ax^2 + bx^1 + cx^0`
Notice that the first term in the general form has an exponent of '2' on `x`, the second term has an exponent of 1 on `x` and the third term has an exponent of 0 on `x`. Thus, in the general form, the terms are written in descending powers of `x`.

Examples

The following quadratic expressions are in the general form:
  • `x^2 + 2x + 1`
  • `10x^2 + 3x + 4`
  • `4x^2 - 8x + 4`
...while the following quadratic equations are not in the general form:
  • `x + 2x^2 + 5`
  • `1 + 3x + 3x^2`
  • `x^2 + 1 + 2x`

Forms of a Quadratic Expression

A quadratic expression can be written in three different ways:

General Form

`ax^2 + bx + c`

Intercept Form

`a(x - p)(x - q)`

Vertex Form

`a(x - h)^2 + k`
Each of the above forms has its own significance, which is discussed in each form's separate post.

Worksheet: Identifying Quadratic Expressions

Instructions

Click on Show Answer button to show a particular answer. Click on the answer itself to hide it. Use the buttons at the bottom to show/hide all answers
Question
Answer
`x^2 + 3x + 4`
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`x^3 + 3x^2 + 1`
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`x + 1`
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`xy + 2x^2 + y^3`
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`xy^2 + x + 1`
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`x^2 + 2xy + y^2`
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`(x^3)/(x) + 2x + 1`
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Solved Examples: Identifying Quadratic Expressions

Identify whether the following expressions are quadratic or not, and state why or why not they are quadratic:

`x^2 + 2x^3 + 3`

This expression is not quadratic because it contains a term with exponent greater than 2 on the variable `x`.

`x + 2`

This expression is not quadratic because it does not contain a term with exponent '2' on the variable.

`x^2 + 2x + 1`

This expression is quadratic because 
  1. the greatest exponent on `x` is 2
  2. `x` does not have negative exponents or exponents greater than 2
`xy + x^2 + y^2`

The above expression is quadratic because its degree is 2. In other words, it is quadratic because the sum of exponents of `x` and `y` in any single term is never greater than 2.

`x^2y + xy + 1`

This expression is not quadratic because the sum of exponents of `x` and `y` in the first term is 3.

Identifying Quadratic Expressions

Definition of Quadratic Expressions

A quadratic expression is an algebraic expression having a degree of 2.

Identifying One Variable Quadratic Expressions

An algebraic expression in one variable is quadratic when:
  • The variable has an exponent of 2 on it
  • The variable does not have an exponent greater than 2
  • The variable does not have a negative exponent
In simple words the above three statements mean:
  • There is at least one term in the expression with `x^2` in it
  • There is no term in the expression with `x^3` or higher exponents in it
  • The expression does not have `x` in the denominator

Example

The algebraic expression `x^2 + 2x + 3` is quadratic because
  • 'x' has an exponent of '2' on it
  • 'x' does not have an exponent greater than 2 on it, and
  • 'x' has an exponent of 1 in the term '2x' and an exponent of 0 in the term '3'

Counter Examples

  • The expression `x^3 + 3x^2` is not quadratic because 'x' has an exponent greater than 2 in it.
  • The expression `x^2 + 1/x` is not quadratic because 'x' is in the denominator (and hence has negative exponents)
  • The expression `0x^2 + 3x + 2` is not quadratic because there is no term with `x^2` in it (the term `0x^2` equals 0 so it is not considered)

Identifying Two Variable Quadratic Expressions

A quadratic expression can have two variables, given that 
  • the sum of exponents of both variables in any term is never greater than 2, and
  • there is at least one term whose degree is 2
  • no variable is present with a negative exponent

Example

`x^2 + xy + y^2` is a quadratic expression because
  • `x` and `y` do not have exponents greater than 2 in any term
  • sum of exponents of x and y in the term `xy` is 2

Counter Examples

  • `x^2y + xy + 1` is not a quadratic expression because the sum of exponents of `x` and `y` in the term `x^2y` is 3, which is greater than 2
  • `x/y + x^2` is not a quadratic expression because `y` is in the denominator, thus making it a rational expression