Lesson 65: Solving Exponential and Logarithmic Equations
Exponential and logarithmic equations are related. Solving them requires using exponent rules, logarithm rules, and sometimes the inverse relationship between exponentials and logarithms.
Steps to Solve
- For exponential equations: try to express both sides with the same base, then equate exponents.
- If the base cannot be matched, take logarithms of both sides and solve for the variable.
- For logarithmic equations: combine log expressions using the laws of logarithms, then rewrite in exponential form to solve.
- Always check for extraneous solutions when using logarithms.
Examples
Example 1 (Exponential with same base): Solve 2^x = 16
Solution: 2^x = 2^4 → x = 4
Example 2 (Exponential with different base): Solve 3^x = 20
Solution: Take log of both sides → log(3^x) = log(20) → x·log(3) = log(20) → x = log(20)/log(3) ≈ 2.726
Example 3 (Logarithmic equation): Solve log_2(x) = 5
Solution: Convert to exponential → 2^5 = x → x = 32
Example 4 (Combining logs): Solve log(x) + log(x − 3) = 1
Solution: log(x(x − 3)) = 1 → x(x − 3) = 10 → x^2 − 3x − 10 = 0 → x = 5 (discard x = −2)
Practice Problems
- 2^(x+1) = 32
- 5^(2x) = 125
- log_3(x) = 4
- log(x) + log(x+2) = 2
- e^(2x) = 7