Question

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

Answer 1

Answer:

$x=\frac{6\ln \left(2\right)}{2\ln \left(5\right)-\ln \left(2\right)}\\x \approx 1.65$

Step-by-step 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 $\left(x+6\right)\ln \left(2\right) = \ln \left(2\right)x+6\ln \left(2\right)$

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

$\ln \left(2\right)x=2x\ln \left(5\right)-6\ln \left(2\right)\\\ln \left(2\right)x-2x\ln \left(5\right)=-6\ln \left(2\right)\\\left(\ln \left(2\right)-2\ln \left(5\right)\right)x=-6\ln \left(2\right)\\\\x=\frac{6\ln \left(2\right)}{2\ln \left(5\right)-\ln \left(2\right)}\approx 1.65$