Chapter 8: Problem 16

Simplify (a) \(\sqrt{\mathrm{e}^{2 \ln x}}\), (b) \(\log \left(100^{\circ}\right)\).

Short Answer

Expert verified

Question: Simplify the following expressions: a) \(\sqrt{\mathrm{e}^{2 \ln x}}\) b) \(\log\left(100^{\circ}\right)\) Answer: a) x b) \(\log\left(\frac{5\pi}{9}\right)\)

Step by step solution

Convert Nested Exponentials to a Single Exponential

The first step is to combine the exponentials using the property \(a^{\ln(b)} = b^{\ln(a)}\) to simplify the expression. This gives: \[\sqrt{\mathrm{e}^{2 \ln x}} = \sqrt{x^{2 \ln(\mathrm{e})}}\]

Apply the Square Root

Now apply the square root to the expression using the property \(\sqrt{a^b} = a^{\frac{b}{2}}\), which gives: \[\sqrt{x^{2 \ln(\mathrm{e})}} = x^{\frac{2 \ln(\mathrm{e})}{2}}\]

Simplify Exponent

Now simplify the exponent by dividing it by 2: \[x^{\frac{2 \ln(\mathrm{e})}{2}} = x^{\ln(\mathrm{e})}\]

Final Simplification

Finally, recall that \(\ln(\mathrm{e}) = 1\), so the expression simplifies to: \[x^{\ln(\mathrm{e})} = x^1 = x\] So the simplified expression for part (a) is x. For part (b), follow these steps:

Convert Degrees to Radians

In this problem, we need to convert 100 degrees to radians to solve the logarithm. To do this, we'll use the conversion factor \(\frac {\pi}{180}\): \[100^{\circ} = 100 \times \frac{\pi}{180} = \frac{5\pi}{9}\]

Apply the Logarithm

Now that we have converted the angle to radians, we can find the logarithm of the value: \[\log\left(100^{\circ}\right) = \log\left(\frac{5\pi}{9}\right)\] Since there are no further simplifications possible for this expression, the simplified expression for part (b) is \(\log\left(\frac{5\pi}{9}\right)\).

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Simplify (a) \(\sqrt{\mathrm{e}^{2 \ln x}}\), (b) \(\log \left(100^{\circ}\right)\).

Short Answer

Step by step solution

Convert Nested Exponentials to a Single Exponential

Apply the Square Root

Simplify Exponent

Final Simplification

Convert Degrees to Radians

Apply the Logarithm

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