decimal help

Question

how do I solve 2^x = 1000

 

(solve for x (round to 2 decimal places)

 

thanks

Answer 1

How do I solve 2^x = 1000?

 

2^x = 10³

 

log₂(2^x) = log₂(10³)        [Take log base 2 of both sides]

 

xlog₂(2) = 3log₂(10)       [Log identity: log(a^b) = b*log(a)]

 

x = 3log₂(10)                 [log₂(2) = 1]

 

x = 9.97                         [log₂(10) = 3.32; 3.32*3 = 9.97]

 

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Alternative - Take log₁₀ of both sides vice log₂:

 

2^x = 10³

 

xlog₁₀(2) = 3log₁₀(10)

 

x = 3/log₁₀(2)                    [Log₁₀(10) = 1]

 

x = 3/.301 = 9.97