Determine the following (if necessary use ^ to indicate exponents, e.g. 2^(16) indicates $2^{16}$):
", "advice": "a)
\n$\\begin{align*}\\log_b\\left(a^\\var{power}\\right) &= \\var{power}\\log_b\\left(a\\right)&& \\text{(by the log law }\\log_b(a^n)=n\\log_b(a)\\text{)}\\\\ &=\\var{power}\\times \\var{c}&& \\text{(given)}\\\\&=\\var{power*c} \\end{align*}$
\n\nb)
\n$\\begin{align*}\\var{mult}\\log_b\\left(a\\right) &=\\log_b\\left(a^\\var{mult}\\right)&& \\text{(by the log law }\\log_b(a^n)=n\\log_b(a)\\text{)} \\end{align*}$
\n\nc)
\n$\\begin{align*}\\log_b\\left(\\sqrt[\\var{root}]{x}\\right) &=\\log_b\\left(x^{\\frac{1}{\\var{root}}}\\right)&& \\text{(by the index law }\\sqrt[n]{x}=x^{\\frac{1}{n}}\\text{)}\\\\[3pt] &=\\frac{1}{\\var{root}}\\log_b(x)&& \\text{(by the log law }\\log_b(a^n)=n\\log_b(a)\\text{)}\\\\[3pt]&=\\frac{1}{\\var{root}}\\times \\var{d} &&\\text{(given)}\\\\[3pt]&=\\simplify{{d}/{root}}\\end{align*}$
\n\nd)
\n$\\begin{align*}\\frac{1}{2}\\log_b\\left(\\var{square}\\right)&=\\log_b\\left(\\var{square}^{\\frac{1}{2}}\\right) &&\\text{(by the log law }\\log_b(a^n)=n\\log_b(a)\\text{)} \\\\&=\\log_b\\left(\\sqrt{\\var{square}}\\right)&& \\text{(by the index law }\\sqrt[n]{x}=x^{\\frac{1}{n}}\\text{)}\\\\[3pt] &=\\log_b(\\var{ans4})\\end{align*}$
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