91Ó°ÊÓ

Exponential equations are equations in which variables appear as exponents. These equations take the form of \(b^y = x\), where \(b\) is the base, \(y\) is the exponent, and \(x\) is the resultant number. To solve exponential equations like \(b^{18} = 64\), we often need to isolate the base and exponent using various mathematical properties.
A common method to solve exponential equations is to use logarithms. This involves converting the equation from its exponential form to a logarithmic form, which can simplify solving for the base or exponent. Remember:

• If \(b^y = x\), then \(\log_b x = y\). This allows you to change the perspective and solve for the unknown.

It’s crucial to understand this conversion as it forms the basis for solving many logarithmic and exponential equations by interchanging forms.

The change of base formula is a powerful tool that allows us to evaluate logarithms with bases other than 10 or \(e\) by converting them into a more manageable form. Despite its absence in solving the given exercise directly, understanding this formula helps solve more complex logarithms.
This formula states:
\[ \log_b x = \frac{\log_c x}{\log_c b} \]
Here, \(c\) can be any positive base not equal to 1. Most commonly, we use base 10 (common logarithm) or base \(e\) (natural logarithm) for calculations. This transforms complex logarithmic problems into simpler ones that can be solved using a calculator or log tables. Knowing how to workaround different bases is useful in both academic problems and real-world applications.

Understanding the properties of exponents is essential for solving equations like \(b^{18} = 64\). These properties help simplify expressions and solve exponential equations effectively. Here are some key properties:

• Power of a Power: \((a^m)^n = a^{m\cdot n}\)
• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Root and Power Rule: \(\sqrt[n]{a^m} = a^{m/n}\)

In the given solution, the root and power rule is particularly useful. It converts roots into fractional exponents, making it possible to simplify \( \sqrt[18]{2^6} \) to \(2^{6/18}\). This simplification is critical in finding \(b = 2^{1/3}\), which is much easier to interpret and use.