Solving Exponential Equations

Solving Exponential Equations with the Same Base

• To solve an exponential equation, aim to express both sides with the same base.
• If bases are equal, then exponents must also be equal.
• For example, in 3^(x+2) = 9^(2x-3), convert 9 to 3^2, then equate the exponents.
• When raising one exponent to another, multiply them (e.g., (3^2)^(2x-3) becomes 3^(4x-6)).

Techniques for Changing Bases

• Convert numbers to a common base to solve exponential equations.
• Recognizing common bases is crucial (e.g., 8 and 16 can both be expressed as powers of 2).
• After establishing a common base, set the exponents equal to each other.

Solving Exponential Equations by Converting to a Common Base (Example)

• Starting equation: 8^(4x-12) = 16^(5x-3)
• Convert 8 to 2^3 and 16 to 2^4: (2^3)^(4x-12) = (2^4)^(5x-3)
• Multiply exponents: 2^(12x-36) = 2^(20x-12)
• Set exponents equal: 12x - 36 = 20x - 12
• Solve for x: x = -3

Common Bases

• 27 and 81 can both be expressed as powers of 3.

Solving Exponential Equations Using Logarithms

• Logarithms can be used to solve exponential equations when you can't easily manipulate to get the same base
• Apply log or natural log (ln) to both sides of the equation

Using Logarithms to Solve for x

• In the equation 3^x = 8, take the log of both sides: log(3^x) = log(8).
• Use the power rule to move the exponent: x * log(3) = log(8).
• Isolate x: x = log(8) / log(3).
• Calculate x: x ≈ 1.8928.

Natural Logarithms

• Natural logarithms (ln) are particularly useful when dealing with the base e.
• If e^x = 7, take the natural log of both sides: ln(e^x) = ln(7).
• Simplify: x * ln(e) = ln(7), where ln(e) = 1, so x = ln(7).
• Decimal approximation: x ≈ 1.9459.

Isolating Exponential Terms Before Applying Logarithms

• Simplify the equation by isolating the exponential term.
• Apply logarithms after isolating the exponential term.
• Solve for x after applying logarithms and using log properties.
• In the equation 5 + 4^(x-2) = 23, subtract 5 from both sides to get 4^(x-2) = 18.
• Apply log to both sides: log(4^(x-2)) = log(18).
• Move the exponent: (x - 2) * log(4) = log(18).
• Isolate x: x = 2 + log(18) / log(4).
• Calculate x: x ≈ 4.08496.

Solving Exponential Equations with Base e

• For equations with base e, use natural logarithms (ln).
• Before taking the natural log, isolate the exponential term.
• Use properties of logarithms to solve for x.
• If 3 + 2e^(3-x) = 7, isolate the exponential term to get 2e^(3-x) = 4.
• Divide by 2: e^(3-x) = 2.
• Apply ln to both sides: ln(e^(3-x)) = ln(2).
• Simplify: 3 - x = ln(2).
• Solve for x: x = 3 - ln(2).
• Approximation: x ≈ 2.3069.

Exponential Equation with Variable in the Exponent and a Fraction

• Rewrite fractions with negative exponents to match bases.
• Set the exponents equal after establishing common bases.
• Solve for x after equating the exponents.
• If 3^(x^2 + 4x) = 1/27, rewrite as 3^(x^2 + 4x) = 3^(-3).
• Then, x^2 + 4x = -3.
• Rewrite as x^2 + 4x + 3 = 0.
• Factor: (x + 3)(x + 1) = 0.
• Solutions: x = -3, x = -1

Solving Exponential Equations with Combined Terms

• When multiplying terms with the same base, add the exponents.
• Factor the resulting quadratic equation to find solutions for x.
• Check solutions by substituting back into original equation.
• For 2^(x^2) * 2^(3x) = 16, rewrite as 2^(x^2 + 3x) = 2^4.
• Then, x^2 + 3x = 4.
• Rewrite as x^2 + 3x - 4 = 0.
• Factor: (x + 4)(x - 1) = 0.
• Solutions: x = -4, x = 1

Substitution to Simplify Factoring

• Use substitution to reduce to a quadratic equation when direct factoring isn't obvious.
• Factor the quadratic equation and solve for the substitute variable.
• Substitute back to find values for the original variable x.
• For 4^(2x) - 20 * 4^x + 64 = 0, let a = 4^x, then the equation becomes a^2 - 20a + 64 = 0.
• Factor: (a - 16)(a - 4) = 0.
• Solutions: a = 16, a = 4.
• If a = 16, 4^x = 16, so x = 2.
• If a = 4, 4^x = 4, so x = 1

Factoring Out the Greatest Common Factor (GCF)

• When terms have a common exponential expression, factor out the GCF.
• Simplify and solve for x.
• For 3^(2x) - 3^(2x-1) = 18, factor out 3^(2x) to get 3^(2x) * (1 - 3^(-1)) = 18.
• Simplify: 3^(2x) * (1 - 1/3) = 18.
• Further simplification: 3^(2x) * (2/3) = 18.
• Multiply by 3/2: 3^(2x) = 27.
• Rewrite 27 as 3^3 so, 3^(2x) = 3^3, thus 2x = 3.
• Solve for x: x = 3/2.