How do you solve for x in ${log}_{2} x - {log}_{2} 3 = 5$ $log_2x-log_2 3=5$ ?

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Answer 1 · 2017-02-01T09:09:04

$x = 96$ $x=96$

Let's first combine the left side, using the rule that ${log}_{x} (a) - {log}_{x} (b) = {log}_{x} (\frac{a}{b})$ $log_x(a)-log_x(b)=log_x(a/b)$ :

${log}_{2} (x) - {log}_{2} (3) = 5$ $log_2(x)-log_2(3)=5$

${log}_{2} (\frac{x}{3}) = 5$ $log_2(x/3)=5$

We can now set these as exponents to a base of 2 (to eliminate the log):

$2^{{log}_{2} (\frac{x}{3})} = 2^{5}$ $2^(log_2(x/3))=2^5$

On the left side, this has the effect of taking the inverse function of the log and so cancels out, leaving:

$\frac{x}{3} = 32$ $x/3=32$

$x = 96$ $x=96$

And now let's check it:

${log}_{2} (96) - {log}_{2} (3) = 5 \Rightarrow 6.585 - 1.585 = 5$ $log_2(96)-log_2(3)=5=>6.585-1.585=5$

(the logs are rounded to 3 decimal places)

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