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Pure · Exponentials & logarithms

Chapter 1 · 3

The idea

Solving exponential and logarithmic equations

Taking logs to solve a^x = b, equations with e^x and ln, hidden quadratics in e^x or a^x, and log equations solved with the laws — always checking which roots are valid.

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Pure · Exponentials & logarithms

Solving exponential and logarithmic equations

Taking logs to solve a^x = b, equations with e^x and ln, hidden quadratics in e^x or a^x, and log equations solved with the laws — always checking which roots are valid.

Why it works

Take logs to free the exponent

The logarithms lesson ended with a tease: 2x=1002^x = 100 finally had an exact name, x=log⁡2100x = \log_2 100 — but naming isn't solving, and nothing back then could touch e2x−5ex+6=0e^{2x} - 5e^x + 6 = 0 or an equation with the unknown inside a log. This is the lesson where all of them fall. First, the basic move: when the unknown is stuck in the exponent, take logs of both sides and use the power law to bring it down. To solve ax=ba^x = b:

ax=b  ⇒  log⁡ax=log⁡b  ⇒  xlog⁡a=log⁡b  ⇒  x=log⁡blog⁡a.a^x = b \;\Rightarrow\; \log a^x = \log b \;\Rightarrow\; x\log a = \log b \;\Rightarrow\; x = \frac{\log b}{\log a}.

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