Mathematics
Exponential Equations
An exponential equation has the variable in the exponent (e.g. 2^x = 32). When the bases match, equate exponents directly. When they don't, take a logarithm of both sides — that brings the variable out of the exponent.
How to Approach Exponential Equations
Try matching bases first
Rewrite both sides with the same base if possible. 2^x = 32 → 2^x = 2^5 → x = 5. Fast and clean when it works.
Otherwise, take a logarithm
Apply log (or ln) to both sides. log(b^x) = log(c) → x·log(b) = log(c) → x = log(c)/log(b). Works for any positive base.
Solve and check
After isolating x, plug back into the original equation. Exponential growth makes errors compound — verification catches them.
Frequently Asked Questions
What if there are two exponential terms?+
Try substitution. For 2^(2x) - 5·2^x + 6 = 0, let u = 2^x. Then u² - 5u + 6 = 0, a regular quadratic. Solve for u, then back-substitute.
Why are exponentials everywhere?+
They model anything that grows or decays at a rate proportional to the current amount — populations, radioactive decay, compound interest, viral spread.
How is e different from other bases?+
e is the unique base where the exponential is its own derivative: d/dx(e^x) = e^x. Calculus is dramatically cleaner with base e.
Related Topics
More step-by-step guides in Mathematics and adjacent subjects.
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