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Logarithms are a way to express exponents. They help us understand how many times one number (the base) needs to be multiplied by itself to produce another number. The most commonly used logarithms are base 10 (common logarithms) and base e (natural logarithms).

When you see \(\text{log}_b(a)\), it means 'to what power must the base \(b\) be raised to get \(a?\).' For example, \(\text{log}_2(8) = 3\) because \(2^3=8\). Natural logarithms, denoted as \(\text{ln}\), use the mathematical constant \(e\) as the base.

Example: \(\text{ln}(e^2) = 2\) because raising the base \(e\) to the power 2 results in \(e^2\). In the given exercise, we simplified \(\text{ln}(e^{6x})\) to \(6x\) using the properties of logarithms.

Understanding the properties of logarithms is essential for solving logarithmic equations. Here are some useful properties:

• Product Property: \(\text{log}_b(MN) = \text{log}_b(M) + \text{log}_b(N)\)
  
• Quotient Property: \(\text{log}_b\bigg(\frac{M}{N}\bigg) = \text{log}_b(M) - \text{log}_b(N)\)
  
• Power Property: \(\text{log}_b(M^p) = p \text{log}_b(M)\)
  

For natural logarithms:

• \text{ln}(e^x) = x
• \text{ln}(1) = 0
• \text{ln}(ab) = \text{ln}(a) + \text{ln}(b)
• \text{ln}\bigg(\frac{a}{b}\bigg) = \text{ln}(a) - \text{ln}(b)

In our exercise, we used the property \(\text{ln}(e^y) = y\), which allows us to simplify \(\text{ln}(e^{6x})\) to \(6x\).

To solve equations, especially logarithmic equations, you need to isolate the variable.

Here are some common steps:

• Step 1: Simplify the equation using logarithmic properties. For example, we simplified \(\text{ln}(e^{6x}) = 18\) to \(6x = 18\).
  
• Step 2: Isolate the variable by performing algebraic operations. In this case, divide both sides by 6 to get \(x = 3\).
  
• Step 3: Check your solution by plugging it back into the original equation. For example, if \(x = 3\), then \(e^{6x} \) should equal \(e^{18}\). Since \(\text{ln}(e^{18}) = 18\), our solution is correct.
  

Always keep these steps in mind: simplify, isolate the variable, and check your work. This approach ensures you solve equations accurately and efficiently.