Solve (a) \(\mathrm{e}^{x}=5\) (b) \(\mathrm{e}^{x}=0.5\) (c) \(\mathrm{e}^{x}=25\) (d) \(\mathrm{e}^{x}=0.001761\)

The natural logarithm, denoted as \(\ln\), is a mathematical function that is the inverse of the exponential function with base \(e\). It is 'natural' because the base \(e\) (approximately 2.71828) is a fundamental constant in mathematics, often arising in calculus and complex analyses.

When dealing with exponential equations like \(e^x = 5\), taking the natural logarithm of both sides allows us to isolate \(x\) because of the inverse relationship between \(e^x\) and \(\ln(x)\). The equation \(\ln(e^x)\) simplifies to just \(x\), since the logarithm asks the question: 'To what power must \(e\) be raised to obtain \(x\)?' In the context of our problems, applying \(\ln\) to both sides gives us the answers directly because \(\ln(e^x) = x\).

Exponential functions follow several important properties that are crucial when solving equations involving exponents. Some properties partake directly in simplifying exponential equations:

• \(e^x \cdot e^y = e^{x+y}\)
• \(\frac{e^x}{e^y} = e^{x-y}\)
• \((e^x)^y = e^{xy}\)
• \(e^0 = 1\)

Using these properties strategically can simplify complex equations or isolate the variable of interest, as seen when we take the natural logarithm of an exponential function to solve for \(x\). For instance, if we had an equation like \(e^{2x} = e^3\), we could simply equate the exponents according to the property that if \(e^a = e^b\), then \(a = b\).

Logarithmic equations contain an unknown variable within a logarithm. Solving these equations often involves the use of logarithmic properties and understanding the relationship between logarithms and exponential functions.

Key properties include:

• \(\log_b(mn) = \log_b(m) + \log_b(n)\)
• \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\)
• \(\log_b(m^n) = n \cdot \log_b(m)\)
• For \(b^m = n\), it follows that \(m = \log_b(n)\)

In our exercise, \(\ln\) is used to solve for \(x\) in the equation \(e^x = a\). By taking the natural log of both sides, we're essentially asking for the power that \(e\) must be raised to in order to achieve \(a\), which directly yields the solution \(x = \ln(a)\). This process is much more intuitive than trying to guess and check what power of \(e\) will give us the desired outcome.