Question

52.Consider the functionsf(x) = 2* x+6 and g(x) = 5*2x.

a. Use properties of logarithms to solve the equation f(x)= g(x). Give your answer as a logarithmic expression, and approximate it to two decimal places.

Answer 1

`x=(6\ln \left(2\right))/(2\ln \left(5\right)-\ln \left(2\right))\\x \approx 1.65`

Explanation:

Given

`f(x) = 2^(\left(x+6\right))`

`g(x) = 5^(2x)`

We want to find
`f(x) = g(x)`

For this you need to:

if
`f(x) = g(x)`, then
`\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)`

`\ln \left(2^(x+6)\right)=\ln \left(5^(2x)\right)`

Apply log rule:
`\log _a\left(x^b\right)=b\cdot \log _a\left(x\right)`

`\left(x+6\right)\ln \left(2\right)=2x\ln \left(5\right)`

Solve for x

Expand
`\left(x+6\right)\ln \left(2\right) = \ln \left(2\right)x+6\ln \left(2\right)`

`\ln \left(2\right)x+6\ln \left(2\right)=2x\ln \left(5\right)`

`\ln \left(2\right)x=2x\ln \left(5\right)-6\ln \left(2\right)\\\ln \left(2\right)x-2x\ln \left(5\right)=-6\ln \left(2\right)\\\left(\ln \left(2\right)-2\ln \left(5\right)\right)x=-6\ln \left(2\right)\\\\x=(6\ln \left(2\right))/(2\ln \left(5\right)-\ln \left(2\right))\approx 1.65`