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Isolating the exponential gives \(\displaystyle 6^{x}=48\). Taking the logarithm of both sides gives \(\displaystyle x = \log_{6}(48)\)

Download \(\LaTeX\)

\begin{question}Solve $6^{x} + 27=75$. 
    \soln{9cm}{Isolating the exponential gives $6^{x}=48$. Taking the logarithm of both sides gives $x = \log_{6}(48)$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}

\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}

\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
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\everymath{\displaystyle}
\ShowSolutiontrue

\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

Solve <img class="equation_image" title=" \displaystyle 6^{x} + 27=75 " src="/equation_images/%20%5Cdisplaystyle%206%5E%7Bx%7D%20%2B%2027%3D75%20" alt="LaTeX: \displaystyle 6^{x} + 27=75 " data-equation-content=" \displaystyle 6^{x} + 27=75 " /> .

Isolating the exponential gives <img class="equation_image" title=" \displaystyle 6^{x}=48 " src="/equation_images/%20%5Cdisplaystyle%206%5E%7Bx%7D%3D48%20" alt="LaTeX: \displaystyle 6^{x}=48 " data-equation-content=" \displaystyle 6^{x}=48 " /> . Taking the logarithm of both sides gives <img class="equation_image" title=" \displaystyle x = \log_{6}(48) " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20%5Clog_%7B6%7D%2848%29%20" alt="LaTeX: \displaystyle x = \log_{6}(48) " data-equation-content=" \displaystyle x = \log_{6}(48) " />