Solving Logarithmic Equations

Solving Basic Logarithmic Equations

• To solve log base 2 of 16 = x, convert it to exponential form: 2^x = 16
• Since 2^4 = 16, then x = 4.
• A change of base formula can be used: log(16) / log(2) = 4 = x

Solving Logarithmic Equations with Unknown Base

• Given log base x of 81 = 4, convert to exponential form: x^4 = 81.
• x is the fourth root of 81/ x = 3 because 3^4 = 81.

Solving Logarithmic Equations with Unknown Argument

• For log base 5 of x = 3, convert to exponential form: 5^3 = x.
• x = 125.

Solving Logarithmic Equations with Fractional Exponents

• For log base 32 of x = 4/5, convert to exponential form: 32^(4/5) = x.
• Simplify: Find the fifth root of 32, then raise to the fourth power.
• Since 2^5 = 32, then 32^(1/5) = 2.
• 2^4 = 16, so x = 16.

Solving Logarithmic Equations with Algebraic Expressions

• Given log base 3 of (5x + 1) = 4, convert to exponential form: 3^4 = 5x + 1.
• 3^4 = 81, therefore 81 = 5x + 1.
• Solving for x: 80 = 5x, so x = 16.

Solving Logarithmic Equations with No Base Specified

• If log x = 24, the base is assumed to be 10, so 10^24 = x.

Solving Natural Logarithmic Equations

• If ln x = 7, the base is e (Euler's number), so e^7 = x.
• x ≈ 1096.65.

Solving Logarithmic Equations Involving Quadratic Expressions

• Given log base 7 of (x^2 + 3x + 9) = 2, convert to exponential form: 7^2 = x^2 + 3x + 9.
• 7^2 = 49, therefore 49 = x^2 + 3x + 9.
• Simplify to quadratic form: x^2 + 3x - 40 = 0.
• Factor the quadratic equation: (x + 8)(x - 5) = 0.
• Therefore, x = -8 and x = 5.

Solving Natural Logarithmic Equations with Algebraic Manipulation

• Given ln(3x - 2) = 5, convert to exponential form: e^5 = 3x - 2.
• Solve for x: e^5 + 2 = 3x, so x = (e^5 + 2) / 3.
• x ≈ 50.14.

Solving Multi-Step Natural Logarithmic Equations

• Given 4 * ln(2x - 1) + 3 = 11, isolate the logarithmic term.
• Subtract 3 from both sides: 4 * ln(2x - 1) = 8.
• Divide by 4: ln(2x - 1) = 2.
• Convert to exponential form: e^2 = 2x - 1.
• Solve for x: x = (e^2 + 1) / 2.
• x ≈ 4.1945.

Solving Equations with Logarithms on Both Sides

• If log base 3(5x + 2) = log base 3(7x - 8), the arguments must be equal.
• Then: 5x + 2 = 7x - 8.
• Solving for x: 10 = 2x, so x = 5.

Solving Equations with Factorable Logarithms

• Given log base 2(x^2 + 4x) = log base 2(5), then x^2 + 4x = 5.
• Rearrange to quadratic form: x^2 + 4x - 5 = 0.
• Factor: (x + 5)(x - 1) = 0.
• Therefore, x = -5 and x = 1.

Combining Logarithms Using the Product Rule

• log a + log b = log (a * b)
• Given log base 2(x) + log base 2(x + 4) = 5, combine the logarithms: log base 2(x * (x + 4)) = 5.
• Convert to exponential form: 2^5 = x(x + 4).
• Simplify: 32 = x^2 + 4x.
• Rearrange to quadratic form: x^2 + 4x - 32 = 0.
• Factor: (x + 8)(x - 4) = 0.
• Potential solutions: x = -8 and x = 4.
• Check for extraneous solutions since logarithms cannot have negative parameters

Extraneous Solutions Example

• Check if x = -8 works: it doesn't, because you can't take the log of a negative number.
• Check if x = 4 works: log base 2(4) + log base 2(4 + 4) = 5.
• log base 2(4) = 2 and log base 2(8) = 3, and since 2 + 3 = 5, x = 4 works.

Solving Equations by Combining Logarithms and Converting to Exponential Form

• Given log base 3(x + 1) = 3 - log base 3(x + 7), move the second logarithm to the left side.
• The equation becomes: log base 3(x + 1) + log base 3(x + 7) = 3.
• Combine the logarithms: log base 3((x + 1)(x + 7)) = 3.
• Convert to exponential form: 3^3 = (x + 1)(x + 7).
• Simplify: 27 = x^2 + 8x + 7.
• Rearrange to quadratic form: x^2 + 8x - 20 = 0.
• Factor: (x + 10)(x - 2) = 0.
• Potential solutions: x = -10 and x = 2.

Identifying Extraneous Solutions

• For log base 3(x + 1) = 3 - log base 3(x + 7), x = -10 is not a valid answer.
• x = 2 works because log base 3(2 + 1) = 3 -log base 3(2 + 7)

Solving Equations with Subtraction Between Two Logs

• Log a - Log b = Log(a / b)

Solving Logarithmic Equations Using Division

• Given log base 4(2x + 6) - log base 4(x - 1) = 1, combine the logarithms: log base 4((2x + 6) / (x - 1)) = 1.
• Convert to exponential form: 4^1 = (2x + 6) / (x - 1).
• Simplify: 4 = (2x + 6) / (x - 1).
• Cross multiply: 4(x - 1) = 2x + 6.
• Solve for x: 4x - 4 = 2x + 6, 2x = 10, x = 5.
• Ensure that x = 5 results in positive values inside the original logistical expressions

Multi-Step Logarithmic Equation Solution Example

• Given log base 2(x + 3) - log base 2(x - 3) = 4, combine the logarithms using division: log base 2((x + 3) / (x - 3)) = 4.
• Convert to exponential form: 2^4 = (x + 3) / (x - 3).
• Simplify: 16 = (x + 3) / (x - 3).
• Cross multiply: 16(x - 3) = x + 3.
• Solve for x: 16x - 48 = x + 3, 15x = 51, x = 51/15, x = 17/5.
• When solving for a variable with logs, you must move constants without variables onto one side

Solving Equations with Logarithms as Exponents

• If log x^(log x) = 49, move the exponent to the front.
• (log x) *(log x) = 49 which is (log x)^2 = 49
• Take the square root of both sides: log x = ±7.
• Convert to exponential form (base 10): x = 10^7 or x = 10^-7.

Solving Different Bases of Log

• log(x^2) = (log x)^2: Convert by changing the base using properties of mathematics; this expression cannot be reduced.
• First move "2" upfront, but only to the left side of the equation: 2log(x) = (log x)^2
• Bring variables to one side: 0 = (log x)^2 -2log(x)
• Factor out a log x: 0 = log(x) (log(x) - 2)
• set each piece equal to zero: log(x) = 0 which is 10^0, thus x = 1
• log(x) - 2 = 0, which is log(x) = 2, which is 10^2, thus x = 100
• check: log((1)^2) = [log(1)]^2, which is 0 = 0
• check second solution: log((100)^2) = [log(100)]^2 = 4 = 4

Nested Logarithms

• When simplifying a logarithmic function that has a nested logarithm, extract the log form outside in.
• For instance: log(log(x)) = 4, thus 10^4 = log(X), which simplifies to 10,000=log(x)
• Simplify: 10^(10,000)=x

How to solve expressions with multiple instances of logs with numbers

• Given Log base 3 of Log base 2 of x, set it all = 2: Log3(Log2(x))=2
• Simplify: 3^2=log2(x) , so 9 = log2(x)
• Further simplify: 2^9=x = 512