Basic logarithmic identity. Properties of logarithms (sum, difference, power)

Basic logarithmic identity

Let $a>0$ , $a\neq1$ , $b>0$ . From $\log_a b=c\ \Leftrightarrow\ a^c=b$ follow two tight identities: $a^{\log_a b}=b$ — the basic logarithmic identity: base $a$ “cancels” $\log_a$ on positive $b$ . Mirror form: $\log_a(a^c)=c$ for any $c\in\mathbb{R}$ (log “strips” the power $a^c$ leaving exponent $c$ ). For $a=10$ : $10^{\lg b}=b$ ; for $a=e$ : $e^{\ln b}=b$ .

The identity $a^{\log_a b}=b$ often collapses powers in algebra

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