Solve Logarithmic Equations  Detailed Solutions
Solve logarithmic equations including some challenging questions. Detailed solutions are presented. The logarithmic equations in examples 4, 5, 6 and 7 involve logarithms with different bases and are therefore challenging.
Example 1: Solve the logarithmic equation
log_{2}(x  1) = 5.
Solution to example 1

Rewrite the logarithm as an exponential using the definition.
x  1 = 2^{5}

Solve the above equation for x.
x = 33

check:
Left Side of equation
log_{2}(x  1) = log_{2}(33  1) = log_{2}(2^{5}) = 5
Right Side of equation = 5

conclusion: The solution to the above equation is
x = 33
Example 2: Solve the logarithmic equation
log_{5}(x  2) + log_{5}(x + 2) = 1.
Solution to example 2

Use the product rule to the expression in the right side.
log_{5}(x  2)(x + 2) = 1

Rewrite the logarithm as an exponential (definition).
(x  2)(x + 2) = 5^{1}

Which can be simplified as.
x^{2} = 9

Solve for x.
x = 3 and x = 3

check:
1st solution x = 3
Left Side of equation: log_{5}(3  2) + log_{5}(3 + 2) = log_{5}1 + log_{5}(3 + 2) = log_{5}5 = 1
Right Side of Equation
2nd solution x = 3
Left Side of equation: log_{5}(3  2) + log_{5}(3 + 2) = log_{5}(5) + log_{5}(1)
log_{5}(5) and log_{5}(1) are both undefined and therefore x = 3 is not a solution.
conclusion: The solutions to the given equation is x = 3
Example 3: Solve the logarithmic equation
log_{3}(x  2) + log_{3}(x  4) = log_{3}(2x^2 + 139)  1.
Solution to example 3

We first replace 1 in the equation by log_{3}(3) and rewrite the equation as follows.
log_{3}(x  2) + log_{3}(x  4) = log_{3}(2x^2 + 139)  log_{3}(3)

We now use the product and quotient rules of the logarithm to rewrite the equation as follows.
log_{3}[ (x  2)(x  4) ] = log_{3}[ (2x^2 + 139) / 3 ]

Which gives the algebraic equation
(x  2)(x  4) = (2x^2 + 139) / 3

Mutliply all terms by 3 and simplify
3(x  2)(x  4) = (2x^2 + 139)

Solve the above quadratic equation to obtain
x = 5 and x = 23

check:
1) x =  5 cannot be a solution to the given equation as it would make the argument of the logarithmic functions on the right negative.
2) x = 23
Right Side of equation:
log_{3}(23  2) + log_{3}(23  4) = log_{3}(21*19) = log_{3}(399)
Left Side of equation:
log_{3}(2(23)^2 + 139)  1 = log_{3}(1197)  log_{3}(3) = log_{3}(1197 / 3) = log_{3}(399)

conclusion: The solution to the above equation is
x = 23
Example 4: Solve the logarithmic equation
log_{4}(x + 1) + log_{16}(x + 1) = log_{4}(8).
Solution to example 4

We first note that 2 logarithms in the given equation have base 4 and one has base 16. We first use the change of base formula to write that
log_{16}(x + 1) = log_{4}(x + 1) / log_{4}(16) = log_{4}(x + 1) / 2 = log_{4}(x + 1)^{1/2}

We now write the given equation as follows.
log_{4}(x + 1) + log_{4}(x + 1)^{1/2} = log_{4}(8)

We use the product rule to write.
log_{4}(x + 1)(x + 1)^{1/2} = log_{4}(8)

Which gives
(x + 1)(x + 1)^{1/2} = (8)

which can be written as
(x + 1)^{3/2} = (8)

Solve for x to obtain.
x = 3

check:
Left Side of equation:
log_{4}(3 + 1) + log_{16}(3 + 1) = 1 + 1/2 = 3/2
Right Side of equation:
log_{4}(8) = log_{4}(4^{3/2}) = 3/2

conclusion: The solution to the above equation is
x = 3
Example 5: Solve the logarithmic equation
log_{2}(x  4) + log_{sqrt(2)}(x^{3}  2) + log_{0.5}(x  4) = 20.
Solution to example 5

We first use the change of base formula to write.
log_{sqrt(2)}(x^{3}  2) = log_{2}(x^{3}  2) / log_{2}(sqrt(2)) = 2log_{2}(x^{3}  2)

We also use the change of base formula to write.
log_{0.5}(x  4) = log_{2}(x  4)

Substitute into the equation and simplify the given equation.
2 log_{2}(x^{3}  2) = 20

rewrite as.
log_{2}(x^{3}  2) = 10

which gives
x^{3}  2 = 2^{10}

Solve the above equation for x.
x = cube_root (1026)
Example 6: Solve the logarithmic equation
ln(x + 6) + log(x + 6) = 4.
Solution to example 6

Use the change of base formula to rewrite log(x + 6) as
log(x + 6) = ln(x + 6) / ln(10)

and substitute in the given equation
ln(x + 6) + ln(x + 6) / ln(10) = 4

solve for ln(x + 6)
ln(x + 6) = 4 ln(10) / (1 + ln(10))

solve the above for x
x = e^{4 ln(10) / (1 + ln(10))}  6
Example 7: Solve the logarithmic equation
log_{5}(ln(x + 3)  1) + log_{1/5}(ln(x + 3)  1) = 0.
Solution to example 7

The change of base formula is used to write
log_{1/5}(ln(x + 3) = log_{5}(ln(x + 3)

Substitute in the given equation
log_{5}(ln(x + 3)  1)  log_{5}(ln(x + 3)  1) = 0

The left hand term is equal to 0 for x + 3 > 0 and ln(x + 3)  1 > 0.
x + 3 > 0 gives x > 3

ln(x + 3)  1 > 0 gives.
ln(x + 3) > 1

or
x + 3 > e

or
x > e  3

conclusion: The solution set to the above equation is given by the interval (e  3 , + infinity). It is an identity.
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Updated: February 2015
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