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maxonik [38]
3 years ago
10

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

Mathematics
1 answer:
likoan [24]3 years ago
8 0

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

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