Exponential Equations

**magentarita** · August 10th 2008, 06:31 AM

Solve each equation below. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places.

(1) (cuberoot{2})^(2 - x) = 2^(x^2)

(2) e^(x + 3) = pi^(x)

**skeeter** · August 10th 2008, 06:56 AM

$(\sqrt[3]{2})^{2-x} = 2^{x^2}$

$(\sqrt[3]{2})^{2-x} = [(\sqrt[3]{2})^3]^{x^2}$

$(\sqrt[3]{2})^{2-x} = (\sqrt[3]{2})^{3x^2}$

so ... what can you say about $2-x$ and $3x^2$ ?

$e^{x+3} = \pi^x$

$\ln(e^{x+3}) = \ln(\pi^x)$

$x+3 = x\ln{\pi}$

$3 = x\ln{\pi} - x$

$3 = x(\ln{\pi} - 1)$

$\frac{3}{\ln{\pi} - 1} = x$

**magentarita** · August 10th 2008, 11:23 AM

This is much better than the other reply.

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