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Solving Logarithmic Functions – Explanation & Examples

In this article, we will learn how to evaluate and solve logarithmic functions with unknown variables.

Logarithms and exponents are two topics in mathematics that are closely related, therefore it is useful we take a brief review of exponents.

Exponent is a form of writing the repeated multiplication of a number by itself. An exponential function is of the form f (x) = b y, where b > 0 < x and b ≠ 1. The quantity x is the number, b is the base and y is the exponent or power.

For example, 32 = 2 × 2 × 2 × 2 × 2 = 22.

The exponential function 22 is read as “two raised by the exponent of five” or “two raised to power five” or “two raised to the fifth power.

On the other hand, the logarithmic function is defined as the inverse function of exponentiation. Consider again the exponential function f(x) = by, where b > 0 < x and b ≠ 1. This function can be represented in logarithmic form as.

y = log b x

Then the logarithmic function is given by;

f(x) = log b x = y, where b is the base, y is the exponent and x is the argument.

The function f (x) = log b x is read as “log base b of x.” Logarithms are useful in mathematics because they enable us to perform calculations with very large numbers.

How to Solve Logarithmic Functions?

To solve the logarithmic functions, it is important to make use of exponential functions in the given expression. The natural log or ln is the inverse of e. That means one can undo the other one i.e.

ln (e x) = x

e ln x = x

To solve an equation with logarithm(s), it is important to know their properties.

Properties of logarithmic functions

Properties of logarithmic functions are simply the rules for simplifying logarithms when the inputs are in the form of division, multiplication or exponents of logarithmic values.

Some of the properties are listed below.

  • Product rule

The product rule of logarithm states the logarithm of the product of two numbers having common base is equal to the sum of individual logarithms.

⟹ log a (p q) = log a p + log a q.

  • Quotient rule

The quotient rule of logarithms states that the logarithm of ratio of the two numbers with same bases is equal to the difference of each logarithm.

⟹ log a (p/q) = log a p – log a q

  • Power rule

The power rule of logarithm states that the logarithm of a number with a rational exponent is equal to the product of exponent and its logarithm.

⟹ log a (p q) = q log a p

  • Change of Base rule

⟹ log a p = log x p ⋅ log a x

⟹ log q p = log x p / log x q

  • Zero Exponent Rule

⟹ log p 1 = 0.

Other properties of logarithmic functions include:

  • The bases of an exponential function and its equivalent logarithmic function are equal.
  • The logarithms of a positive number to the base of the same number is equal to 1.

log a a = 1

  • Logarithms of 1 to any base is 0.

log a 1 = 0

  • Log a0 is undefined
  • Logarithms of negative numbers are undefined.
  • The base of logarithms can never be negative or 1.
  • A logarithmic function with base 10is called a common logarithm. Always assume a base of 10 when solving with logarithmic functions without a small subscript for the base.

Comparison of exponential function and logarithmic function

Whenever you see logarithms in the equation, you always think of how to undo the logarithm in order to solve the equation. For that you use exponential function. Both of these functions are interchangeable.

The following table tells the way of writing and interchanging the exponential functions and logarithmic functions. The third column tells about how to read both the logarithmic functions.

Exponential functionLogarithmic functionRead as
82 = 64log 8 64 = 2log base 8 of 64
103 = 1000log 1000 = 3log base 10 of 1000
100 = 1log 1 = 0log base 10 of 1
252 = 625log 25 625 = 2log base 25 of 625
122 = 144log 12 144 = 2log base 12 of 144

Let’s use these properties to solve a couple of problems involving logarithmic functions.

Example 1

Rewrite exponential function 72 = 49 to its equivalent logarithmic function.

Solution

Given 72 = 64.

Here, the base = 7, exponent = 2 and the argument = 49. Therefore, 72 = 64 in logarithmic function is;

⟹ log 7 49 = 2

Example 2

Write the logarithmic equivalent of 53 = 125.

Solution

Base = 5;

exponent = 3;

and argument = 125

53 = 125 ⟹ log 5 125 =3

Example 3

Solve for x in log 3 x = 2

Solution

log 3 x = 2
32 = x
⟹ x = 9

Example 4

If 2 log x = 4 log 3, then find the value of ‘x’.

Solution

2 log x = 4 log 3

Divide each side by 2.

log x = (4 log 3) / 2

log x = 2 log 3

log x = log 32

log x = log 9

x = 9

Example 5

Find the logarithm of 1024 to the base 2.

Solution

1024 = 210

log 2 1024 = 10

Example 6

Find the value of x in log 2 (x) = 4

Solution

Rewrite the logarithmic function log 2(x) = 4 to exponential form.

24 = x

16 = x

Example 7

Solve for x in the following logarithmic function log 2 (x – 1) = 5.

Solution
Rewrite the logarithm in exponential form as;

log 2 (x – 1) = 5 ⟹ x – 1 = 25

Now, solve for x in the algebraic equation.
⟹ x – 1 = 32
x = 33

Example 8

Find the value of x in log x 900 = 2.

Solution

Write the logarithm in exponential form as;

x2 = 900

Find the square root of both sides of the equation to get;

x = -30 and 30

But since, the base of logarithms can never be negative or 1, therefore, the correct answer is 30.

Example 9

Solve for x given, log x = log 2 + log 5

Solution

Using the product rule Log b (m n) = log b m + log b n we get;

⟹ log 2 + log 5 = log (2 * 5) = Log (10).

Therefore, x = 10.

Example 10

Solve log x (4x – 3) = 2

Solution

Rewrite the logarithm in exponential form to get;

x2 = 4x – 3

Now, solve the quadratic equation.
x2 = 4x – 3
x2 – 4x + 3 = 0
(x -1) (x – 3) = 0

x = 1 or 3

Since the base of a logarithm can never be 1, then the only solution is 3.

Practice Questions

1. Express the following logarithms in exponential form.

a. 1og 26

b. log 9 3

c. log4 1

d. log 66

e. log 825

f. log 3 (-9)

2. Solve for x in each of the following logarithms

a. log 3 (x + 1) = 2

b. log 5 (3x – 8) = 2

c. log (x + 2) + log (x – 1) = 1

d. log x4– log 3 = log(3x2)

3. Find the value of y in each of the following logarithms.

a. log 2 8 = y

b. log 5 1 = y

c. log 4 1/8 = y

d. log y = 100000

4. Solve for xif log x (9/25) = 2.

5. Solve log 2 3 – log 224

6. Find the value of x in the following logarithm log 5 (125x) =4

7. Given, Log 102 = 0.30103, Log 10 3 = 0.47712 and Log 10 7 = 0.84510, solve the following logarithms:

a. log 6

b. log 21

c. log 14

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