# Solve Logarithmic Equations with Solutions for Grade 12

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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.

.

### 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.

.

### 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).

.

### 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 \( f_1(x) = log_{x^2}(\dfrac{16}{25}) + 1 / 2\).

.

### 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.

.

### 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.

.

### Question 7

Solve the equation: 10 log(log(x)) = 1.### Solution

Divide 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.

.

## Links and References

Solve Logarithmic EquationsLogarithmic Functions

High School Maths (Grades 10, 11 and 12) - Free Questions and Problems With Answers

Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers

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