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Olenka [21]
2 years ago
12

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.
Mathematics
1 answer:
Naily [24]2 years ago
8 0

The value of x is -1.596

<em><u>Solution:</u></em>

<em><u>Given equation is:</u></em>

4^{(x+3)} = 7

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

\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)

Therefore,

\ln \left(4^{x+3}\right)=\ln \left(7\right)

\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)

\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)\\

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)}\\\\

\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

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

Thus solution for x is found

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