In the logarithm properties section of the book Calculus for Dummies, there is this property: $$a^{\log_{a}b}=b$$ How can I prove it?
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12Welcome to Mathematics Stack Exchange. How do you define $\log_a b$? – J. W. Tanner Jul 18 '23 at 19:19
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3$f(f^{-1}(x))=x$ – MathFail Jul 18 '23 at 19:20
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@MathFail Ah I see! – MastarCheeze Jul 18 '23 at 19:21
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Possibly useful is my answer to Is there a "Exponential Form" of the "Logarithmic Change of Base"?. – Dave L. Renfro Jul 18 '23 at 19:22
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8This equation literally is the definition of $\log_{a}{b}$. There is no need to prove it at all. – Geoffrey Trang Jul 18 '23 at 19:26
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1By definition of log, we are through. $\blacksquare$ – Nothing special Jul 18 '23 at 19:39
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4Right, there is no need to prove it, but there is a need to prove that the definition of $\log_a b$ is well-defined - that is, the function $f(x)=a^x$ has an inverse. (This wouldn't be true when $a=1,$ so we don't define $\log_1 b.$) – Thomas Andrews Jul 18 '23 at 19:40
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@J.W.Tanner I'm not sure about your question, the book only includes the equation without any further explanation. – MastarCheeze Jul 18 '23 at 19:49
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$$b=b$$ $$\implies\log_a b=\log_a b$$ $$\implies(\log_a b)\log_aa=\log_a b$$ As $\log_yy=1$ for any $y$ in the domain $$\implies\log_aa^{\log_ab}=\log_a b$$ It is a rule of logarithms which I hope you know. Now simply if $\log_mn=\log_mp$ It means $n=p$. Using the same reasoning we can say that $$\boxed{a^{\log_{a}b}=b}$$
