Talk:List of logarithmic identities

I had a problem of the form <math> a^x = b</math>, where I wanted to solve for <math>x</math> in terms of <math>a</math> and <math>b</math>. I think the answer is that <math>x</math> equals the ratio of the two logarithms (common or natural or any other base) of the known quantities: <math> x = \frac{\log_a}{\log_b}</math>. Is this correct? If so, it would be worth spelling out, presumably in the "Changing the base" section, for the benefit of readers looking to solve this type of problem. [[User:JamesMLane|JamesMLane]]<small>&nbsp;[[User_talk:JamesMLane|t]]&nbsp;[[Special:Contributions/JamesMLane|c]]</small> 20:48, 4 July 2025 (UTC)

I had a problem of the form <math> a^x = b</math>, where I wanted to solve for <math>x</math> in terms of <math>a</math> and <math>b</math>. I think the answer is that <math>x</math> equals the ratio of the two logarithms (common or natural or any other base) of the known quantities: <math> x = \frac{\log_a}{\log_b}</math>. Is this correct? If so, it would be worth spelling out, presumably in the "Changing the base" section, for the benefit of readers looking to solve this type of problem. [[User:JamesMLane|JamesMLane]]<small>&nbsp;[[User_talk:JamesMLane|t]]&nbsp;[[Special:Contributions/JamesMLane|c]]</small> 20:48, 4 July 2025 (UTC)

:You’re looking for the identity under the “Trivial identities” header’s “Explanations” subheader, which lists <math display="inline">\log_b(y) = x \iff b^x = y</math>, which substituting for your variables nets <math display="inline">\log_a(b) = x \iff a^x = b</math>. Change-of-base is something different. It basically states you can find the logarithm of any base {{mvar|b}} of {{mvar|a}}, by calculating <math display="inline">\frac{\log_c(a)}{\log_c(b)}</math>, where {{mvar|c}} is any [valid] number. Eg: you can use the natural logarithm (log base [[Euler’s number]]) to calculate the binary logarithm (log base 2): <math display="inline">\log_2(x)=\frac{\log_e(x)}{\log_e(2)}=\frac{\ln(x)}{\ln(2)}</math>. [[User:EmptySora|<span style="color: #300057">Empty</span><span style="color: #00BCF0">Sora_</span>]] ([[User talk:EmptySora|talk]]) 21:15, 4 July 2025 (UTC)