Grades Algebra II & Trigonometry Exponential & Logarithmic FunctionsSolving Exponential & Logarithmic Equations

Solving Exponential & Logarithmic Equations

🔴 Exponential & Logarithmic Functions

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Solving Exponential & Logarithmic Equations

Use logarithms to solve exponential equations and vice versa.

To solve exponential equations where the variable is in the exponent, take the logarithm of both sides. To solve logarithmic equations, exponentiate both sides. Always check solutions for extraneous roots (log arguments must be positive).

Solving Exponential Equations

1Isolate the exponential expression on one side.

2Take the natural log (or log base b) of both sides.

3Use the power rule: ln(bˣ) = x·ln(b).

4Solve for the variable algebraically.

5Use a calculator to get a decimal approximation if needed.

3^x = 50 \implies x\ln 3 = \ln 50 \implies x = \frac{\ln 50}{\ln 3} \approx \frac{3.912}{1.099} \approx 3.56

\log_2(3x - 1) = 4 \implies 3x - 1 = 2^4 = 16 \implies x = \frac{17}{3}

🔍Solving a Logarithmic Equation

Solve ln(x) + ln(x − 2) = ln(8). Combine: ln(x(x−2)) = ln(8) → x² − 2x = 8 → x² − 2x − 8 = 0 → (x−4)(x+2) = 0. x = 4 or x = −2. Reject x = −2 (ln is undefined there). Answer: x = 4.

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Remember This!

When solving log equations, always check that all arguments of logarithms are positive in the original equation — "extraneous solutions" often arise from squaring or multiplying.

✏️ Try It!

Solve: 2^(x+1) = 32

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🔋Log & Exponent Evaluator

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