Chapters

## Exercise 1

Solve the logarithmic equations.

1

2

3

5

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## Exercise 2

Solve the logarithmic simultaneous equations.

1

2

3

## Solution of exercise 1

Solve the logarithmic equations.

Applying the logarithmic power rule here, we will get the following expression:

Write the two terms on the left hand side as a single log function by applying logarithm product rule:

Since both sides of the equation has log functions, so you can write the resultant expression without them like this:

Set the equation equal to 0 by taking on the left hand side of the equation:

The above fraction can be written as:

Either or

Hence, , or

2
Apply the logarithm power rule to write the above expression as:
Apply the logarithm quotient rule on the left hand side of the equation:
Cancel the log functions from both sides of the equation and solve the resultant equation algebraically:
Take 2 to the left hand side of the equation:

If we substitute in the original equation, we will end up taking the log of negative number which is impossible. Hence, this equation has No Solution.

3

By taking the factors from right hand side of the equation to the left hand side and setting the equation to 0, we will get the following expression:

Suppose

By substituting the value in the equation, we will get the following new equation:

We will factor the above equation by expanding it and writing the factors in two pairs like this:

Either or

Hence, t = 1 or t = -2

Remember that we assumed , hence we can say that or

By converting the above values in exponential form, we get the following values of :

and

4

Apply the power rule here to write the equation as follows:

Cancel the log functions on both sides of the equation to get the following algebraic expression:

Use the formula to expand the right hand side of the equation:

5

Take the expression from the denominator on the left hand side to the numerator on the right hand side of the equation:

Apply the logarithm power rule here to get the following equation:

Cancel the log functions from both sides of the equation and solve the resultant equation algebraically:

Find factors of above expression by expanding it:

Hence, or

## Solution of exercise 2

Solve the logarithmic simultaneous equations.

1

Use the logarithm product rule on the left hand side of the equation:
In exponential form, it can be written as:
Substitute this value of in the second equation:
Use the quadratic formula to find the value of :

and Hence, or

If , then

If then

3

We can rewrite the second equation using the exponent product rule:

Suppose and

We will solve this equation through substitution:

Substitute this value of in the second equation:

Put this value of in the first equation to get the value of :

Remember that and

Hence, and

Since, 2 raised to the power 2 is equal to 4, so the value of .

Similarly, 3 raised to the power 3 is equal to 27, so .

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