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How to Solve Logarithmic Equations? Questions with detailed solutions are presented. The following rules and properties of logarithms are used to solve these equations.
Properties of Logarithms
Log_{b}A + Log_{b}B = Log_{b}(A B)
Log_{b}A  Log_{b}B = Log_{b}(A / B)
n Log_{b}A = Log_{b}A^{n}
If Log_{b}A = Log_{b}B, then A = B
Also the graphical approximation to the solutions of each equation of the form f(x) = g(x) are shown as the x coordinates of the x intercepts of the graph of the function h(x) = f(x)  g(x). This is done by first writing the equation to solve with its right side equal to zero and then graphing the left side and locating the x intercepts.
Questions: Solve Logarithmic Equations
Question 1
Solve the equation: log(2x  3) = log(3  x)  2.
Solution
Rewrite the equation with the log terms on one side.
log(2x  3)  log(3  x) =  2
Rewrite the equation substituting  2 by log 10^{2}
log(2x  3)  log(3  x) = log 10^{2}
Use the log rule log A  Log B = log (A/B) to rewrite the equation as
log ((2x  3)/(3  x)) = log 10^{2}
Function log(x) being a one to one function, we can write
(2x  3)/(3  x) = 10^{2}
Solve the above equation
2x  3 = (3  x) / 100
200x  300 = 3  x
201x = 303
x = 303 / 201 = 101 / 67 ≈ 1.51
Check the solution found.
left side: log(2(101/67)  3) = log(1/67) =  log(67)
right side: log(3  101 / 67)  2 = log(100/67)  2 = log(100)  log(67)  2 = 2  log(67)  2 =  log(67)
The given equation has one solution.
x = 101 / 67 ≈ 1.51
The x intercept of the graph of the function q(x) = log(2x  3)  log(3  x) + 2 (the left side of the given equation written with its right side equal to zero) is shown below. Note that the x coordinate of the x intercept is close to the solution obtained analytically above.
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Question 2
Solve the equation: log x  log(x^{2}  1) =  2 log(x  1).
Solution
Use the log rule log A  Log B = log (A/B) to rewrite the left side of the equation as one term and the rule n log(x) = log(x^{n}) to rewrite the right side as the log of a power.
log (x/(x^{2}  1)) = log (x  1)^{2}
Function log(x) being a one to one function, we can write
x / (x^{2}  1) = (x  1)^{2}
Multiply all terms of the above equation by (x  1)^{2} and simplify
(x  1)^{2} (x /(x^{2}  1)) = (x  1)^{2} (x  1)^{2}
(x  1)^{2} (x /(x^{2}  1)) = 1
Expand (x  1)^{2} and (x^{2}  1)) and simplify
x(x  1)(x  1) / ((x + 1)(x  1)) = 1
x(x  1) / (x + 1) = 1
Multiply both sides of the equation by x + 1 and simplify.
x(x  1) = x + 1
x^{2}  2 x  1 = 0
Two solutions: x_{1} = 1 + √ 2 ≈ 2.41 and x_{2} = 1  √ 2 ≈  0.41
Check the solutions found.
x_{1} = 1 + √ 2
left side: log (1 + √ 2)  log((1 + √ 2)^{2}  1) = log (1 + √ 2)  log( 2 + 2 √ 2) =  log(2)
Right side:  2 log(1 + √ 2  1) = 2 log(√ 2) =  log (2)
x_{2} = 1  √ 2
Left side: log (1  √ 2)  log((1  √ 2)^{2}  1) is undefined because 1  √ 2 is negative and the term log (1  √ 2) is undefined.
The given equation has one solution.
x = 1 + √ 2 ≈ 2.41
The x intercept of the function r(x) = log x  log(x^{2}  1) + 2 log(x  1) is shown below and its x coordinate is close to the soltuion of the equation.
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Question 3
Solve the equation: log_{2}(2x  9) = 2  log_{2}(x  1).
Solution
Rewrite the equation with terms with log on the same side and substitute 2 by log_{2}4
log_{2}(2x  9) + log_{2}(x  1) = log_{2}4
Use the rule log_{2}A + log_{2}B = log_{2} (A B) to rewrite the equation as follows
log_{2}( (2x  9)(x  1) ) = log_{2}4
Which gives
(2x  9)(x  1) = 4
Expand and write in standard form.
2 x^{2}  11 x + 5 = 0
Solve the above quadratic equation to obtain
Two solutions: x_{1} = 1 / 2 and x_{2} = 5
Check the solutions found.
x_{1} = 1 / 2
left side: log_{2}(2(1/2)  9) undefined since the argument of the log is negative.
x_{2} = 5
Left side: log_{2}(2(5)  9) = 0
Left side: 2  log_{2}(5  1) = 2  log_{2} 4 = 0
The given equation has one solution.
x = 5
The graphical solution is shown below as the x intercept of the function s(x) = log_{2}(2x  9)  2 + log_{2}(x  1).
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Question 4
Solve the equation: $$ log_{x^2}(\dfrac{16}{25}) =  1 / 2.$$
Solution
Use the inverse relationship between the exponential and logarithmic to the same base to rewrite the equation as:
(x^{2})^{1/2} = 16 / 25
Note that (x^{2})^{1/2} = 1 / (x^{2})^{1/2} = 1 /  x  and rewrite the equation as:
1 /  x  = 16 / 25 or  x  = 25 / 16
Which gives the solutions
Two solutions: x_{1} = 25/16 and x_{2} = 25/16
Check the solutions found.
x_{1} = 25/16
left side: log_{(25/16)2}(16/25) = log_{(25/16)2}(25/16)^{1} = log_{(25/16)2}((25/16)^{2})^{1/2} = log_{(25/16)2}((25/16)^{ 2})^{1/2} =  1 / 2
x_{2} =  25/16
left side: log_{(  25/16)2}(16/25) = log_{(25/16)2}(16/25) =  1 / 2
The given equation has two solution.
x = 25 / 16 and x =  25 / 16
The graphical solution is shown below as the x intercepts of \( f_1(x) = log_{x^2}(\dfrac{16}{25}) + 1 / 2\).
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Question 5
Solve the equation: 2 ln(x + 3)  ln(x + 1) = 3 ln 2
Solution
Use the rules n ln x = ln x^{n} and ln (A/B) = ln A  ln B to rewrite the equation as follows:
ln(x + 3)^{2}  ln(x + 1) = ln 2^{3}
ln ((x + 3)^{2} / (x + 1)) = ln 8
ln x is a one to one function, hence
(x + 3)^{2} (x + 1) = 8 or (x + 3)^{2} = 8(x + 1)
Write the above quadratic equation in standard form.
x^{2}  2 x + 1 = 0
Which gives one solution
One solution: x = 1
Check the solutions found.
left side: 2 ln(1 + 3)  ln(1 + 1) = 2 ln 4  ln 2 = 4 ln 2  ln 2 = 3 ln 2
The graphical solution is shown below as the x intercept of h_{1}(x) = 2 ln(x + 3)  ln(x + 1)  3 ln 2.
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Question 6
Solve the equation: (log_{2}(x))^{2}  Log_{2}(x^{2}) = 8.
Solution
Use the rules n Log_{2} x = Log_{2} x^{n} to rewrite the equation as follows:
(log_{2}(x))^{2}  2 Log_{2} (x) = 8
Let u = log_{2}(x) and write the equation in standard form nd in terms of u.
u^{2}  2 u  8 = 0
Which gives two solutions
One solution: u = 2 and u = 4
We now solve for x.
u = =  2 = log_{2}(x) gives x = 2^{2} = 1/4
u = = 4 = log_{2}(x) gives x = 2^{4} = 16
The graphical solutions are shown below as the x intercepts of p_{1}(x) = (log_{2}(x))^{2}  Log_{2}(x^{2}) = 8.
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Question 7
Solve the equation: 10 log(log(x)) = 1.
Solution
Divide both sides by 10
log(log(x)) = 0.1
Which gives
log(x) = 10^{0.1}
Which gives x as
x = 10^{(100.1)} ≈ 18.15
We now solve for x.
The graphical solution is shown below of the x intercepts of f(x) = 10 log(log(x))  1.
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