# How to Solve Logarithmic Equations with Detailed Solutions for Grade 12

| 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 LogarithmsLog_{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.
## Question 1Solve the equation: log(2x - 3) = log(3 - x) - 2.## SolutionRewrite 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. .
## Question 2Solve the equation: log x - log(x^{2} - 1) = - 2 log(x - 1).
## SolutionUse 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 3Solve the equation: log_{2}(2x - 9) = 2 - log_{2}(x - 1).
## SolutionRewrite 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 4Solve the equation: $$ log_{x^2}(\dfrac{16}{25}) = - 1 / 2.$$## SolutionUse 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 \( f_1(x) = log_{x^2}(\dfrac{16}{25}) + 1 / 2\). .
## Question 5Solve the equation: 2 ln(x + 3) - ln(x + 1) = 3 ln 2## SolutionUse 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 6Solve the equation: (log_{2}(x))^{2} - Log_{2}(x^{2}) = 8.
## SolutionUse 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 7Solve the equation: 10 log(log(x)) = 1.## SolutionDivide both sides by 10log(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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Solve Logarithmic EquationsLogarithmic Functions

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