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How do you solve ${log}_{8} 5 - {log}_{8} (x - 6) = 1$ $log_8 5 -log_8(x-6)=1$ ?

1 Answer

Sep 21, 2016

$x = \frac{53}{8} or 6 \frac{5}{8}$ $x=53/8 or 6 5/8$

Explanation:

Remembering that
$XXX {log}_{b} (p) - {log}_{b} (q) = {log}_{b} (\frac{p}{q})$ $color(white)("XXX")color(blue)(log_b(p) - log_b(q)=log_b (p/q))$
and that
$XXX {log}_{b} (b) = 1$ $color(white)("XXX"color(magenta)(log_b(b) = 1)$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

${log}_{8} 5 - {log}_{8} (x - 6) = 1$ $color(blue)(log_8 5 - log_8 (x-6))=color(magenta)(1)$

can be rewritten as
$XXX {log}_{8} (\frac{5}{x - 6}) = {log}_{8} 8$ $color(white)("XXX")color(blue)(log_8(5/(x-6))) =color(magenta)( log_8 8)$

which implies
$XXX \frac{5}{x - 6} = 8$ $color(white)("XXX")5/(x-6)=8$

$XXX \to 5 = 8 x - 48$ $color(white)("XXX")rarr5=8x-48$

$XXX \to 8 x = 53$ $color(white)("XXX")rarr 8x=53$

$XXX \to x = \frac{53}{8}$ $color(white)("XXX")rarr x=53/8$

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