Exercise 1
Solve the logarithmic equations.
1 
2 
3 
4 
5 
Exercise 2
Solve the logarithmic simultaneous equations.
1 
2 
3 
Solution of exercise 1
Solve the logarithmic equations.
1 
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 
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 
in the second equation:
:
2 
Cancel the log function from both sides of the equation:
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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