Explain how solve 4^(x+3)=7 using the change of base formula log base b of y equals log y over log b. Include the solution for x
in your answer. Round your answer to the nearest thousandth.
1 answer:
The value of x is -1.596
<em><u>Solution:</u></em>
<em><u>Given equation is:</u></em>
![4^{(x+3)} = 7](https://tex.z-dn.net/?f=4%5E%7B%28x%2B3%29%7D%20%3D%207)
Let us solve using change of base formula log base b of y equals log y over log b
From given,
![4^{(x+3)} = 7](https://tex.z-dn.net/?f=4%5E%7B%28x%2B3%29%7D%20%3D%207)
![\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)](https://tex.z-dn.net/?f=%5Cmathrm%7BIf%5C%3A%7Df%5Cleft%28x%5Cright%29%3Dg%5Cleft%28x%5Cright%29%5Cmathrm%7B%2C%5C%3Athen%5C%3A%7D%5Cln%20%5Cleft%28f%5Cleft%28x%5Cright%29%5Cright%29%3D%5Cln%20%5Cleft%28g%5Cleft%28x%5Cright%29%5Cright%29)
Therefore,
![\ln \left(4^{x+3}\right)=\ln \left(7\right)](https://tex.z-dn.net/?f=%5Cln%20%5Cleft%284%5E%7Bx%2B3%7D%5Cright%29%3D%5Cln%20%5Cleft%287%5Cright%29)
![\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)](https://tex.z-dn.net/?f=%5Cmathrm%7BApply%5C%3Alog%5C%3Arule%7D%3A%5Cquad%20%5Clog%20_a%5Cleft%28x%5Eb%5Cright%29%3Db%5Ccdot%20%5Clog%20_a%5Cleft%28x%5Cright%29)
![\ln \left(4^{x+3}\right)=\left(x+3\right)\ln \left(4\right)\\\\\left(x+3\right)\ln \left(4\right)=\ln \left(7\right)\\](https://tex.z-dn.net/?f=%5Cln%20%5Cleft%284%5E%7Bx%2B3%7D%5Cright%29%3D%5Cleft%28x%2B3%5Cright%29%5Cln%20%5Cleft%284%5Cright%29%5C%5C%5C%5C%5Cleft%28x%2B3%5Cright%29%5Cln%20%5Cleft%284%5Cright%29%3D%5Cln%20%5Cleft%287%5Cright%29%5C%5C)
Let us simplify the above
![\left(x+3\right)\cdot \:2\ln \left(2\right)=\ln \left(7\right)\\\\\mathrm{Divide\:both\:sides\:by\:}2\ln \left(2\right)\\\\\frac{\left(x+3\right)\cdot \:2\ln \left(2\right)}{2\ln \left(2\right)}=\frac{\ln \left(7\right)}{2\ln \left(2\right)}\\\\](https://tex.z-dn.net/?f=%5Cleft%28x%2B3%5Cright%29%5Ccdot%20%5C%3A2%5Cln%20%5Cleft%282%5Cright%29%3D%5Cln%20%5Cleft%287%5Cright%29%5C%5C%5C%5C%5Cmathrm%7BDivide%5C%3Aboth%5C%3Asides%5C%3Aby%5C%3A%7D2%5Cln%20%5Cleft%282%5Cright%29%5C%5C%5C%5C%5Cfrac%7B%5Cleft%28x%2B3%5Cright%29%5Ccdot%20%5C%3A2%5Cln%20%5Cleft%282%5Cright%29%7D%7B2%5Cln%20%5Cleft%282%5Cright%29%7D%3D%5Cfrac%7B%5Cln%20%5Cleft%287%5Cright%29%7D%7B2%5Cln%20%5Cleft%282%5Cright%29%7D%5C%5C%5C%5C)
![\mathrm{Simplify}\\\\x+3=\frac{\ln \left(7\right)}{2\ln \left(2\right)}\\\\\mathrm{Subtract\:}3\mathrm{\:from\:both\:sides}\\\\x+3-3=\frac{\ln \left(7\right)}{2\ln \left(2\right)}-3\\\\\mathrm{Simplify}\\\\x=\frac{\ln \left(7\right)}{2\ln \left(2\right)}-3](https://tex.z-dn.net/?f=%5Cmathrm%7BSimplify%7D%5C%5C%5C%5Cx%2B3%3D%5Cfrac%7B%5Cln%20%5Cleft%287%5Cright%29%7D%7B2%5Cln%20%5Cleft%282%5Cright%29%7D%5C%5C%5C%5C%5Cmathrm%7BSubtract%5C%3A%7D3%5Cmathrm%7B%5C%3Afrom%5C%3Aboth%5C%3Asides%7D%5C%5C%5C%5Cx%2B3-3%3D%5Cfrac%7B%5Cln%20%5Cleft%287%5Cright%29%7D%7B2%5Cln%20%5Cleft%282%5Cright%29%7D-3%5C%5C%5C%5C%5Cmathrm%7BSimplify%7D%5C%5C%5C%5Cx%3D%5Cfrac%7B%5Cln%20%5Cleft%287%5Cright%29%7D%7B2%5Cln%20%5Cleft%282%5Cright%29%7D-3)
Substitute the values
ln 7 = 1.9459101490553132
ln 2 = 0.6931471805599453
Therefore,
![x = \frac{1.9459101490553132}{2 \times 0.6931471805599453} - 3\\\\x = 1.40367746103 - 3\\\\x = -1.59632253897 \approx -1.596](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B1.9459101490553132%7D%7B2%20%5Ctimes%200.6931471805599453%7D%20-%203%5C%5C%5C%5Cx%20%3D%201.40367746103%20-%203%5C%5C%5C%5Cx%20%3D%20-1.59632253897%20%5Capprox%20-1.596)
Thus solution for x is found
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Step-by-step explanation:
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<h2>SOLUTION :-</h2>
- PLEASE CHECK THE ATTACHED FILE
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Step-by-step explanation: