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Question Number 166619 by BagusSetyoWibowo last updated on 23/Feb/22

(2/(log_(10) (6)))=(π^x /7)  How much the x is?

$$\frac{\mathrm{2}}{\mathrm{log}_{\mathrm{10}} \left(\mathrm{6}\right)}=\frac{\pi^{{x}} }{\mathrm{7}} \\ $$$${How}\:{much}\:{the}\:{x}\:{is}? \\ $$

Answered by TheSupreme last updated on 23/Feb/22

((log_(10) (100))/(log_0 (6)))=log_6 (100)=(π^x /7)  π^x =7log_6 (100)  x=log_π (7log_6 (100))

$$\frac{{log}_{\mathrm{10}} \left(\mathrm{100}\right)}{{log}_{\mathrm{0}} \left(\mathrm{6}\right)}={log}_{\mathrm{6}} \left(\mathrm{100}\right)=\frac{\pi^{{x}} }{\mathrm{7}} \\ $$$$\pi^{{x}} =\mathrm{7}{log}_{\mathrm{6}} \left(\mathrm{100}\right) \\ $$$${x}={log}_{\pi} \left(\mathrm{7}{log}_{\mathrm{6}} \left(\mathrm{100}\right)\right) \\ $$

Commented by BagusSetyoWibowo last updated on 24/Feb/22

Another Method  (2/(log_(10) (6)))=(π^x /7)  Cross multiply if (a/b)=(c/d)  a×d=b×c  2×7=log_(10) (6)×π^x   14=log_(10) (6)×π^x   Divide both sides by log_(10) (6)  ((14)/(log_(10) (6)))=((log_(10) (6)π^x )/(log_(10) (6)))  Switch sides  π^x =((14)/(log_(10) (6)))  Apply exponent rule  xln(π)=ln(((14)/(log_(10) (6))))  Solve  x=((ln(((14)/(log_(10) (6)))))/(ln(π)))  x=2,524518...

$${Another}\:{Method} \\ $$$$\frac{\mathrm{2}}{\mathrm{log}_{\mathrm{10}} \left(\mathrm{6}\right)}=\frac{\pi^{{x}} }{\mathrm{7}} \\ $$$${Cross}\:{multiply}\:{if}\:\frac{{a}}{{b}}=\frac{{c}}{{d}} \\ $$$${a}×{d}={b}×{c} \\ $$$$\mathrm{2}×\mathrm{7}=\mathrm{log}_{\mathrm{10}} \left(\mathrm{6}\right)×\pi^{{x}} \\ $$$$\mathrm{14}=\mathrm{log}_{\mathrm{10}} \left(\mathrm{6}\right)×\pi^{{x}} \\ $$$${Divide}\:{both}\:{sides}\:{by}\:\mathrm{log}_{\mathrm{10}} \left(\mathrm{6}\right) \\ $$$$\frac{\mathrm{14}}{\mathrm{log}_{\mathrm{10}} \left(\mathrm{6}\right)}=\frac{\mathrm{log}_{\mathrm{10}} \left(\mathrm{6}\right)\pi^{{x}} }{\mathrm{log}_{\mathrm{10}} \left(\mathrm{6}\right)} \\ $$$${Switch}\:{sides} \\ $$$$\pi^{{x}} =\frac{\mathrm{14}}{\mathrm{log}_{\mathrm{10}} \left(\mathrm{6}\right)} \\ $$$${Apply}\:{exponent}\:{rule} \\ $$$${x}\mathrm{ln}\left(\pi\right)=\mathrm{ln}\left(\frac{\mathrm{14}}{\mathrm{log}_{\mathrm{10}} \left(\mathrm{6}\right)}\right) \\ $$$${Solve} \\ $$$${x}=\frac{\mathrm{ln}\left(\frac{\mathrm{14}}{\mathrm{log}_{\mathrm{10}} \left(\mathrm{6}\right)}\right)}{\mathrm{ln}\left(\pi\right)} \\ $$$${x}=\mathrm{2},\mathrm{524518}... \\ $$$$ \\ $$

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