Question

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

Short answer

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 \(\log\left(\frac{5\pi}{9}\right)\).

Key concepts

Nested Exponentials

The concept of nested exponentials involves dealing with expressions where an exponent is itself an exponential expression. Let's take a closer look at the original exercise:

• The task is to simplify expressions like \( \sqrt{\mathrm{e}^{2 \ln x}} \). This can seem tricky at first glance, primarily because of the double-layered nature of the exponentials involved.
• A handy property for simplifying nested exponentials is \( a^{\ln(b)} = b^{\ln(a)} \). This property allows us to reconfigure the expression into a more manageable form.

Use this step to "flatten" the exponential before applying other mathematical operations. This method is particularly useful as it prepares the expression for subsequent simplifications. Being comfortable with these properties and their adjustments will greatly aid in tackling complex exponential problems.

Square Root Property

Simplifying expressions with square roots can be made easier with certain mathematical properties. When dealing with an expression like \( \sqrt{x^{2 \ln(\mathrm{e})}} \), you can use the fundamental square root property: \( \sqrt{a^b} = a^{\frac{b}{2}} \).

• By applying this property, you convert the square root into a more straightforward exponential form: an exponent divided by 2.
• This drastically reduces the complexity of the expression, allowing for further simplification.

Understanding this property is key to breaking down complex expressions involving square roots, especially when paired with exponentials. It transforms a potentially intimidating problem into something much simpler.

Degree to Radian Conversion

This conversion process is essential in simplifying expressions involving angles since trigonometric functions and certain logarithmic calculations often require radian measure. The conversion factor used is \( \frac{\pi}{180} \), which translates degrees into radians.

• For instance, to convert 100 degrees to radians, multiply: \( 100 \times \frac{\pi}{180} = \frac{5\pi}{9} \).
• Understanding and correctly applying this factor is crucial for transitioning smoothly between angle units in problems involving angular measurements.

Once your angle is in radians, it can be used seamlessly in various mathematical contexts, particularly when dealing with logarithmic expressions, as seen in part (b) of the original exercise.

Logarithm Properties

Logarithms can transform complex multiplicative relationships into simpler additive ones, making them powerful tools in simplification.When faced with expressions like \( \log\left(\frac{5\pi}{9}\right) \), knowing your logarithm properties can make all the difference:

• Properties such as the product, quotient, and power rules are particularly useful.
• For example, \( \log(ab) = \log a + \log b \) or \( \log\left(\frac{a}{b}\right) = \log a - \log b \).
• These allow decomposing or rearranging expressions into more manageable parts.

Familiarity with these properties enhances your ability to manipulate and simplify logarithmic expressions effectively, supporting the final steps in solutions where logarithms are involved.