Question

Andre says that 1.5" align="absmiddle" class="latex-formula"> because 55 is halfway between 10 and 100. Do you agree with Andre? Explain your reasoning.
I know that answer is false, but I don't know how to explain!!

Answer 1

$\textit{exponential form of a logarithm} \\\\ \log_a(b)=y \qquad \implies \qquad a^y= b \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{lllllllll} \log_{\underline{10}}(10)=1&\implies &\underline{10}^1=10 ~~ \checkmark\\\\ \log_{10}(100)=2&\implies &10^2=100~~ \checkmark\\\\ \log_{10}(55)=1.5&\implies &10^{1.5}=55\implies 10^{\frac{3}{2}}=55\implies \sqrt[2]{10^3}=55\\\\ &&\sqrt{1000}\ne 55 ~~ \bigotimes \end{array}$

increments over the exponent, are not directly proportional to increments on the resulting values, namely the half-way from 10² and 10⁴ is not 10³, because 10⁴ is 100 times 10², and 10³ is simply 10 times 10².