Answer:
[tex]x=\frac{6\ln \left(2\right)}{2\ln \left(5\right)-\ln \left(2\right)}\\x \approx 1.65[/tex]
Step-by-step explanation:
Given
[tex]f(x) = 2^{\left(x+6\right)}[/tex]
[tex]g(x) = 5^{2x}[/tex]
We want to find [tex]f(x) = g(x)[/tex]
For this you need to:
if [tex]f(x) = g(x)[/tex], then [tex]\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)[/tex]
[tex]\ln \left(2^{x+6}\right)=\ln \left(5^{2x}\right)[/tex]
Apply log rule: [tex]\log _a\left(x^b\right)=b\cdot \log _a\left(x\right)[/tex]
[tex]\left(x+6\right)\ln \left(2\right)=2x\ln \left(5\right)[/tex]
Solve for x
Expand [tex]\left(x+6\right)\ln \left(2\right) = \ln \left(2\right)x+6\ln \left(2\right)[/tex]
[tex]\ln \left(2\right)x+6\ln \left(2\right)=2x\ln \left(5\right)[/tex]
[tex]\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[/tex]
