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How do you calculate ${log}_{343} 49$ $log_(343) 49$ ?

1 Answer

May 4, 2016

${log}_{343} 49 = \frac{2}{3}$ $log_343 49 = 2/3$ (method below)

Explanation:

Notice that
$XXX 49 = 7^{2}$ $color(white)("XXX")49=7^2$
and
$XXX 343 = 7^{3}$ $color(white)("XXX")343=7^3$

and remember that ${log}_{b} a = k$ $log_b a = k$ is equivalent to $b^{k} = a$ $b^k=a$

Let ${log}_{343} 49 = k$ $color(blue)(log_343 49=k)$

$XXX \Rightarrow 343^{k} = 49$ $color(white)("XXX")rArr 343^k=49$

$XXX \Rightarrow {(7^{3})}^{k} = 7^{2}$ $color(white)("XXX")rArr (7^3)^k=7^2$

$XXX \Rightarrow 7^{3 k} = 7^{2}$ $color(white)("XXX")rArr 7^(3k)=7^2$

$XXX \Rightarrow 3 k = 2$ $color(white)("XXX")rArr 3k=2$

$XXX \Rightarrow k = \frac{2}{3}$ $color(white)("XXX")rArr color(blue)(k=2/3)$

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