Solving Exponential and Logarithmic Equations: Method Selection and Domain Checks

Exponential and logarithmic equations become much easier when the method is chosen from the equation’s structure. Some equations should be rewritten with a common base; some need logarithms; some need substitution; and logarithmic equations always need domain checks.

1. Common base first

Solve 2^(x+1)=16. Since 16=2⁴, equate exponents: x+1=4, so x=3.

Taking logarithms would work, but it adds unnecessary steps.

2. Use logarithms when a common base is not convenient

Solve 3^x=7. Taking natural logarithms gives xln3=ln7, so x=ln7/ln3.

3. Isolate the exponential expression

Solve 5(2^x)−3=17. First isolate: 5(2^x)=20, so 2^x=4. Hence x=2.

4. Substitution for repeated exponential structure

Solve 4^x−5(2^x)+4=0. Let u=2^x, so 4^x=(2^x)²=u².

Then u²−5u+4=0, giving u=1 or 4. Therefore 2^x=1 or 4, so x=0 or 2.

5. Logarithmic equations need positive arguments

Solve ln(x−1)=ln5. The domain requires x>1. Since ln is one-to-one on positive inputs, x−1=5, giving x=6.

6. Combine logs only when valid

Solve ln x+ln(x−2)=ln8. Require x>2. Combine: ln[x(x−2)]=ln8, so x²−2x=8.

x²−2x−8=0 gives x=4 or −2. The domain removes −2. Therefore x=4.

7. A log equation can produce an algebraic candidate outside the domain

Domain checking is part of the solution, not an optional verification. A candidate making any original logarithm argument nonpositive must be rejected.

8. Exact answers before decimals

For 5^x=11, x=ln11/ln5 is exact. Approximate only if requested. Keeping the quotient of logarithms preserves the relationship and avoids premature rounding.

9. Common mistakes

• Taking log of only one side of an equation.
• Using log(a+b)=loga+logb.
• Forgetting that u=2^x must be positive after substitution.
• Keeping a root that violates a logarithm domain.

10. Practice

1. Solve 3^(x−1)=27.
2. Solve 2^x=10 exactly.
3. Solve 9^x−10(3^x)+9=0.
4. Solve ln(x+2)=ln7.
5. Solve ln x+ln(x−3)=ln4.

11. Answers

1. x−1=3, so x=4.
2. x=ln10/ln2.
3. Let u=3^x>0. u²−10u+9=0 gives u=1 or 9, so x=0 or 2.
4. x=5.
5. Require x>3. x(x−3)=4 gives x=4 or −1; only x=4 works.

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