Unit 2 · Topic 2.13

2.13 Exponential and Logarithmic Equations and Inequalities

To solve an exponential equation, isolate the power and take a log. To solve a log equation, combine logs and rewrite in exponential form, then throw out any answers outside the domain. The same moves find inverses of exponential and log functions.

Key terms

exponential equation
logarithmic equation
extraneous solution
inequality
inverse

Solving exponential equations

Isolate the exponential part, then use the definition of a log. For 4 · 3^(2x) = 100: divide by 4 to get 3^(2x) = 25, rewrite as 2x = log₃ 25, and solve: x = (1/2)log₃ 25 = log₃ 5.

If both sides can be written with the same base, compare exponents instead: 4ˣ = 8^(x − 1) becomes 2^(2x) = 2^(3x − 3), so 2x = 3x − 3 and x = 3.

Taking ln or log of both sides also works: 3^(2x) = 25 gives 2x ln 3 = ln 25.

Solving log equations

Use the log properties to combine everything into one log, then rewrite in exponential form. log_b(A) = c becomes A = bᶜ.

Then check every answer in the original equation. A value that makes any log's input zero or negative is an extraneous solution and must be rejected.

Inequalities

Solve exponential and log inequalities the same way, but watch the direction. Taking a log, or exponentiating, keeps the inequality's direction when the base is greater than 1 and reverses it when the base is between 0 and 1.

Example: (0.5)ˣ < 0.1. Since (0.5)ˣ is decreasing, this happens for x greater than log₀.₅(0.1) = log₂(10) ≈ 3.322. So x > log₂ 10.

For log inequalities, also include the domain condition: log₂(x − 1) < 3 means 0 < x − 1 < 8, so 1 < x < 9.

Rewriting with a different base

Any exponential can be rewritten with another base: bˣ = c^(x · log_c b). For example, 2ˣ = e^(x ln 2). This is how models written with e connect to models written with other bases.

Inverses of transformed exponentials and logs

For f(x) = a · b^(x − h) + k or f(x) = a · log_b(x − h) + k, find the inverse by undoing the steps in reverse order: subtract k, divide by a, take the log (or exponentiate), then add h.

The inverse of an exponential is a log, and the inverse of a log is an exponential. The domain of the inverse is the range of the original.

Worked examples

Try each one yourself first, then open the solution.

Example 1
An exponential equation
Solve 4 · 3^(2x) = 100 exactly, then approximate.
Show the solutionHide the solution
1. Step 1: Divide by 4: 3^(2x) = 25.
2. Step 2: Rewrite in log form: 2x = log₃ 25.
3. Step 3: x = (1/2) log₃ 25 = log₃(25^(1/2)) = log₃ 5.
4. Step 4: Approximate: log₃ 5 = ln 5 / ln 3 ≈ 1.465.
Answer: x = log₃ 5 ≈ 1.465
Example 2
Trap: an extraneous solution
Solve log₂(x) + log₂(x − 2) = 3.
Show the solutionHide the solution
1. Step 1: Combine with the product rule: log₂(x(x − 2)) = 3.
2. Step 2: Exponential form: x(x − 2) = 2³ = 8, so x² − 2x − 8 = 0.
3. Step 3: Factor: (x − 4)(x + 2) = 0, so x = 4 or x = −2.
4. Step 4: Check x = −2: log₂(−2) is undefined, so reject it. Check x = 4: log₂ 4 + log₂ 2 = 2 + 1 = 3. It works.
Answer: x = 4 (x = −2 is extraneous).
Example 3
Inverse of a transformed exponential
Find the inverse of f(x) = 3 · 2^(x − 1) + 4 and its domain.
Show the solutionHide the solution
1. Step 1: Write y = 3 · 2^(x − 1) + 4 and swap: x = 3 · 2^(y − 1) + 4.
2. Step 2: Subtract 4 and divide by 3: (x − 4)/3 = 2^(y − 1).
3. Step 3: Log form: y − 1 = log₂((x − 4)/3), so y = 1 + log₂((x − 4)/3).
4. Step 4: f's range is y > 4 (its asymptote is y = 4), so the inverse's domain is x > 4.
Answer: f⁻¹(x) = 1 + log₂((x − 4)/3), for x > 4.

Common mistakes

Not isolating the exponential first. In 4 · 3^(2x) = 100, taking the log before dividing by 4 leads to errors like log(12^(2x)).
Keeping extraneous solutions. Always check that every log's input is positive.
Forgetting to flip an inequality when the base is between 0 and 1.
Writing log₂(x) + log₂(x − 2) as log₂(2x − 2). The product rule multiplies the inputs.

On the exam

The symbolic-manipulation free-response question regularly includes solving an exponential or log equation for exact values. Show the rewrite into log or exponential form, and state why you reject any extraneous answer.
Exact answers like log₃ 5 or (ln 7)/2 are usually expected on no-calculator parts; don't convert to decimals unless asked.

Connected topics

Videos

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