Lesson 2-3: Solving Exponential & Logarithmic Equations

Learning Objectives

Use same-base transformations to equate exponents and solve.
Apply natural logs or common logs to linearize exponentials.
Perform substitution (e.g., t=2^x) to reduce to quadratics.

Solve log equations and check positivity of all arguments.
Interpret solutions in modeling contexts and reject extraneous roots.

Solution Strategies

Same-Base Method

Rewrite both sides as a^f(x) and a^g(x); if a>0,a≠1 then f(x)=g(x).

Take Logs

Apply ln or log to both sides; bring exponents down via power rule.

Substitution

Let t=a^x or t=b^x to convert to polynomial form; solve then back-substitute.

Domain Check

Ensure log arguments >0 and denominators ≠0; discard invalid candidates.

Worked Examples

Example 1: Same-Base Method

Solve $2^{x+3}=8^2$ .

Show Solution

$8=2^3$ ⇒ $8^2=2^6$ . Equate exponents: $x+3=6$ ⇒ $x=3$ .

Example 2: Taking Logs

Solve $3^x=7$ for x.

Show Solution

Take ln: $x=\log_3 7=\tfrac{\ln 7}{\ln 3}$ ≈ 1.771.

Example 3: Substitution t=2^x

Solve $4^x - 10\cdot 2^x + 16 = 0$ .

Show Solution

Let $t=2^x>0$ , then $t^2 - 10t + 16=0$ ⇒ $(t-2)(t-8)=0$ ⇒ $t=2$ or $t=8$ ⇒ $x=1$ or $x=3$ .

Example 4: Log Equation with Domain Check

Solve $\log_2(x+1)+\log_2(x-2)=\log_2(4x-5)$ .

Show Solution

Combine LHS: $\log_2[(x+1)(x-2)]$ ⇒ set arguments equal: $(x+1)(x-2)=4x-5$ ⇒ $x^2 - x - 2=4x-5$ ⇒ $x^2 - 5x + 3=0$ ⇒ $x=\tfrac{5\pm\sqrt{13}}{2}$ . Check domain: need $x>2$ and $4x-5>0$ ; only $x=\tfrac{5+\sqrt{13}}{2}$ valid.

Example 5: Mixed Techniques

Solve $2^{2x+1} = 5\cdot 3^x$ .

Show Solution

Take ln: $(2x+1)\ln 2 = \ln 5 + x\ln 3$ ⇒ $x(2\ln 2 - \ln 3)=\ln 5 - \ln 2$ ⇒ $x=\tfrac{\ln(5/2)}{2\ln 2 - \ln 3}$ .

Practice Problems

Problem 1

Solve $5^{x-2}= frac{1}{25}$ .

Show Solution

$1/25=5^{-2}$ ⇒ $x-2=-2$ ⇒ $x=0$ .

Problem 2

Solve $\ln(x+3)-\ln(x-1)=\ln 2$ (check domain).

Show Solution

Combine: $\ln\left(\tfrac{x+3}{x-1}\right)=\ln 2$ ⇒ $\tfrac{x+3}{x-1}=2$ ⇒ $x+3=2x-2$ ⇒ $x=5$ ; domain x>1, valid.

Problem 3

Solve $3^{2x}-7\cdot 3^x+12=0$ .

Show Solution

Let $t=3^x>0$ , then $t^2-7t+12=0$ ⇒ $(t-3)(t-4)=0$ ⇒ $t=3$ or $t=4$ ⇒ $x=1$ or $x=\log_3 4$ .

Problem 4

Solve $\log_4(x-1)+\log_4(x-3)=2$ .

Show Solution

$\log_4[(x-1)(x-3)]=2$ ⇒ $(x-1)(x-3)=16$ ⇒ $x^2-4x-13=0$ ⇒ $x=2\pm \sqrt{17}$ . Domain requires x>3; valid root: $x=2+\sqrt{17}$ .

Problem 5

Solve $e^{2x}-7e^{x}+10=0$ .

Show Solution

Let $t=e^{x}>0$ , then $t^2-7t+10=0$ ⇒ $(t-5)(t-2)=0$ ⇒ $x=\ln 5$ or $x=\ln 2$ .

Key Takeaways

Convert to same base when feasible; otherwise apply ln/log.
Substitution turns exponential equations into polynomial ones.
Always check domains in log equations to avoid extraneous roots.
Interpret solutions within context; units and feasibility matter.

Advanced Techniques & Pitfalls

Mixed Bases

Use change-of-base inside equations to align bases before solving.

Extraneous Roots

After squaring or combining logs, verify domain to eliminate invalid solutions.

Practice Bank

Bank 1: Same-Base Conversion

Rewrite both sides with a common base.
Equate exponents and solve.
Check numerical validity.

Bank 2: Taking Logs

Apply ln to both sides carefully.
Bring exponents down using power rule.
Solve linear equation in x.

Bank 3: Substitution

Let $t=a^x$ reduce to polynomial.
Solve for t, then back-substitute.
Reject negatives where t>0 required.

Bank 4: Log Equations

Combine logs using identities.
Translate to algebraic equation.
Verify domain of each log argument.

Bank 5: Mixed Bases

Use change-of-base inside equations.
Control rounding error.
Present answers to 3 decimals.

Bank 6: Inequalities

Transform using monotonicity of bases.
Be careful when base in (0,1).
Express solution as intervals.

Bank 7: Parameter Recovery

Use two conditions to solve for a and b.
Check with a third point.
State model explicitly.

Bank 8: Domain Pitfalls

After squaring, re-check positivity.
Eliminate extraneous solutions.
Explain failures succinctly.

Bank 9: Application Word Problems

Translate context to equations.
Define units and parameters.
Interpret solution.

Bank 10: Challenge Mix

Mix same-base, logs, substitution.
Show domain checks clearly.
Provide exact and decimal forms.

FAQ (Extended)

Q: How to start with mixed exponential–log equations?

Combine logs when possible, or take logs to linearize exponentials; use substitution to simplify the structure.

Q: Quick domain checks after solving?

Substitute solutions back into each log’s argument and denominators to ensure positivity and nonzero; a number line helps avoid misses.

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