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Solve the following equation for $n$

$\\displaystyle{\\simplify{{a}={p}({b})^(n/{d})+{c}}}.$

$n=$ [[0]]

Note: Typing $\\log(5)$ will input the value $\\log_{10}(5)$, whereas $\\log5$ will not work.
Note: Typing $\\ln(5)$ will input the value $\\log_e(5)$, whereas $\\ln5$ will not work.

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We start solving the equation one operation at a time by doing the inverse to both sides, when we get to undoing the exponential we apply a log to both sides, we then use a log law and continue solving.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\var{a}$	$=$	$\\simplify{{p}({b})^(n/{d})+{c}}$
$\\simplify{{a-c}}$	$=$	$\\simplify{{p}({b})^(n/{d})}$	(subtract $\\var{c}$ from both sides)
$\\var{frac}$	$=$	$\\simplify{{b}^(n/{d})}$	(divide both sides by $\\var{p}$)
$\\log(\\var{frac})$	$=$	$\\log(\\var{b}^{\\frac{n}{\\var{d}}})$	(take the log of both sides)
	$=$	$\\frac{n}{\\var{d}}\\log(\\var{b})$	(use a log law)

$\\displaystyle{\\frac{\\log(\\var{frac})}{\\log(\\var{b})}}$	$=$	$\\frac{n}{\\var{d}}$	(divide both sides by $\\log(\\var{b})$)

$\\displaystyle{\\frac{\\var{d}\\log(\\var{frac})}{\\log(\\var{b})}}$	$=$	$n$	(multiply both sides by $\\var{d}$)

Solve the following equation for $n$

$\\displaystyle{\\simplify{{FV}={pay}((1+{int})^n-1)/{int}}}.$

$n=$ [[0]]

Note: Typing $\\log(5)$ will input the value $\\log_{10}(5)$, whereas $\\log5$ will not work.
Note: Typing $\\ln(5)$ will input the value $\\log_e(5)$, whereas $\\ln5$ will not work.

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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

${\\var{FV}}$	$=$	$\\displaystyle{\\simplify{{pay}((1+{int})^n-1)/{int}}}$
$\\simplify{{FV*int}}$	$=$	$\\displaystyle{\\simplify{{pay}((1+{int})^n-1)}}$	(multiply both sides by $\\var{int}$)
$\\displaystyle{\\simplify[simplifyFractions]{{FV*int}/{pay}}}$	$=$	$\\displaystyle{\\simplify{(1+{int})^n-1}}$	(divide both sides by $\\var{pay}$)
$\\displaystyle{\\simplify[simplifyFractions]{{FV*int}/{pay}+1}}$	$=$	$\\displaystyle{\\simplify{(1+{int})^n}}$	(add $1$ to both sides)
$\\displaystyle{\\simplify[simplifyFractions]{{FV*int+pay}/{pay}}}$	$=$	$\\displaystyle{\\simplify{(1+{int})^n}}$	(tidy up left hand side)

$\\displaystyle{\\log\\left(\\simplify[simplifyFractions]{{FV*int+pay}/{pay}}\\right)}$	$=$	$\\displaystyle{\\log\\left((\\var{1+int})^n\\right)}$	(take the log of both sides)

$\\displaystyle{\\log\\left(\\simplify[simplifyFractions]{{FV*int+pay}/{pay}}\\right)}$	$=$	$\\displaystyle{n\\log(\\var{1+int})}$	(use a log law)

$\\displaystyle{\\frac{\\log\\left(\\simplify[simplifyFractions]{{FV*int+pay}/{pay}}\\right)}{\\log(\\var{1+int})}}$	$=$	$n$	(divide both sides by $\\log(\\var{1+int})$)

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