Nick88888888
  • Nick88888888
(1.05)^x = 2.5 Round your answer to the nearest tenth. how would i do this step by step? 2. ___ √ 3x = 50 Round your answer to the nearest tenth.
Mathematics
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
Apply logarithm of base 10 to both the sides \[\log_{10}((1.05)^{x})=\log_{10}(2.5)\] Rule: 1.\[\log(x^y)=y.\log(x)\] (true for all base) 2.\[\log_{a}a=1\] log of any number to it's base is 1 3.\[\log(a \times b)=\log(a)+\log(b)\] 4.\[\log(\frac{a}{b})=\log(a)-\log(b)\] Also true for all base 5.\[\log(1)=0\] Also true for all base \[x.\log_{10}(1.05)=\log_{10}(\frac{10}{4})\] Here's the first I've applied property 1 to left side and written right side 2.5 as 10/4 use property 4 on right side now Note on the right side I've written 2.5 as 10/4
Nick88888888
  • Nick88888888
Ok so i just do (1.05)(10/4) or... what
anonymous
  • anonymous
Do you know about logarithms?

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anonymous
  • anonymous
ok Consider an example \[(0.4)^x=\frac{0.25}{100}\] Apply log of base 10 to both sides \[\log_{10}(0.4)^x=\log_{10}(\frac{0.25}{100})\] Use rule 1 on left side and rule 4 on right side \[x.\log_{10}(0.4)=\log_{10}(0.25)-\log_{10}(100)\] Now we can write 0.4 as 4/10 0.25 as square of 0.5 and 100 as square of 10 so do that \[x.\log_{10}(\frac{4}{10})=\log_{10}((0.5)^2)-\log_{10}(10^2)\] Property 4 on left and property property 1 on right \[x[\log_{10}(4)-\log_{10}(10)]=2\log_{10}(0.5)-2\log_{10}(10)\] We can write 4 as 2 square and 0.5 as 1/2 so do that \[x[\log_{10}(2^2)-\log_{10}(10)]=2[\log_{10}(\frac{1}{2})-\log_{10}(10)]\] Use properties to simplify further \[x[2\log_{10}(2)-\log_{10}(10)]=2[\log_{10}(1)-\log_{10}(2)-\log_{10}(10)]\] Now put the values in, \[\log_{10}(2)=0.3010\] By property 2 and property 5 the other values are \[\log_{10}(10)=1\]\[\log_{10}(1)=0\]\[x[2 \times 0.3010-1]=2[0-0.3010-1]\]\[-0.398x=-2.06\]\[x=\frac{2.06}{0.398}\approx 5.176\]
Nick88888888
  • Nick88888888
sorry didnt mean to bump

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