Question

What is the solution to the equation log 2 x – log 2 4 = 5 ?

Answer 1

is the equation like this?
\[\log_2 x - \log_2 4 = 5\]

Answer 2

yes!

Answer 3

alright, there is one property of logs we can use to combine the two logs on the left hand side of the equation:\[\log\left(\frac{x}{y}\right)= \log x- \log y\]

Answer 4

So using this (in the reverse order), we see that:\[\log_2 x - \log_2 4 = \log_2 \left(\frac{x}{4}\right)\]does that make sense?

Answer 5

(not that we are done yet lol)

Answer 6

yes lol

Answer 7

just one more step. So now the equation we have is:\[\log_2 \left(\frac{x}{4}\right)=5\]We are going to use the property:\[\log_b x = y\iff b^y = x\]These two equations are equivalent. So we obtain:\[\log_2 \left(\frac{x}{4}\right) = 5 \iff 2^5 = \frac{x}{4}\] Im sure you can take it from here.

Answer 8

not really.

Answer 9

x=4?

Answer 10

Eek. alright, first we need to calculate the left hand side. This gives:\[2^5 = 2\cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32\]So now the equation is:\[32 = \frac{x}{4}\]

Answer 11

From there, we multiply both sides of the equation by 4, and we get:\[4\cdot 32 = 4\cdot \frac{x}{4} \iff 128 = x\]