Question

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.

Answer 1

To solve the equation 4^(x+3)=7 using the change of base formula, we can take the logarithm of both sides of the equation and apply the change of base formula to isolate x. Finally, calculate the value of x using a calculator and round to the nearest thousandth.

Step-by-step explanation:

To solve the equation 4^(x+3)=7 using the change of base formula, we can use the formula log base b of y equals log y over log b. In this case, we will use the base 10 logarithm.

1. Take the logarithm of both sides of the equation: log(4^(x+3)) = log 7
2. Apply the change of base formula: (x+3)log4 = log 7
3. Divide both sides of the equation by log4 to isolate x+3: (x+3) = log 7 / log 4
4. Subtract 3 from both sides of the equation to solve for x: x = (log 7 / log 4) - 3
5. Calculate the value of x using a calculator and round to the nearest thousandth.

Answer 2

The value of x is -1.596

Solution:

Given equation is:

`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)\\\\(\left(x+3\right)\cdot \:2\ln \left(2\right))/(2\ln \left(2\right))=(\ln \left(7\right))/(2\ln \left(2\right))\\\\`

`\mathrm{Simplify}\\\\x+3=(\ln \left(7\right))/(2\ln \left(2\right))\\\\\mathrm{Subtract\:}3\mathrm{\:from\:both\:sides}\\\\x+3-3=(\ln \left(7\right))/(2\ln \left(2\right))-3\\\\\mathrm{Simplify}\\\\x=(\ln \left(7\right))/(2\ln \left(2\right))-3`

Substitute the values

ln 7 = 1.9459101490553132

ln 2 = 0.6931471805599453

Therefore,

`x = (1.9459101490553132)/(2 * 0.6931471805599453) - 3\\\\x = 1.40367746103 - 3\\\\x = -1.59632253897 \approx -1.596`

Thus solution for x is found