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Exponentiation is a fundamental mathematical operation that involves raising a number, known as the base, to the power of another number, called the exponent. For example, in the expression \( 7^x \), 7 is the base and \( x \) is the exponent. This means that 7 is multiplied by itself \( x \) times. Exponentiation is used in various fields such as algebra, engineering, and computer science due to its ability to represent large numbers compactly.

• Base: The number being multiplied.
• Exponent: Indicates how many times the base is used as a factor.

Understanding exponentiation helps clarify how equations, like \( 7^x = \frac{1}{15} \), can be solved by applying logarithms to isolate the variable, providing a deeper insight into the problem-solving process.

A logarithm is the inverse operation of exponentiation, just like subtraction is the inverse of addition. It answers the question: "To what exponent must the base be raised, to produce a given number?" In the equation \( 7^x = \frac{1}{15} \), to solve for \( x \), you convert it to logarithmic form \( x = \log_7 \left( \frac{1}{15} \right) \). This conversion is crucial because it transforms the multiplication process in exponentiation into a more manageable form for solving. Logarithms have specific properties that allow us to simplify complex expressions:

• Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \).
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \).
• Power Rule: \( \log_b(M^n) = n \cdot \log_b(M) \).

By understanding logarithms, you can solve equations where the variable is an exponent, as done in the example, making seemingly complex calculations straightforward.

The change of base formula is a tool used to convert a logarithm of any base into a different base, typically base-10 or base-e, which are usually available on scientific calculators. It states that for every positive number \( a \), where \( a eq 1 \), and all positive numbers \( M \) and \( N \), \( \log_a(N) = \frac{\log_b(N)}{\log_b(a)} \). This formula is applied to make the computation of \( \log_7 \left( \frac{1}{15} \right) \) more straightforward and practical by changing it into base-10, as shown:

• \( \log_7 \left( \frac{1}{15} \right) = \frac{\log_{10} \left( \frac{1}{15} \right)}{\log_{10}(7)} \)

The change of base formula simplifies the use of logarithms in diverse situations, improving your efficiency in solving logarithmic equations.

Solving equations involves finding the value of the variables that make the equation true. In our case, we need to solve the equation \( 7^x = \frac{1}{15} \) for \( x \). Solving equations requires a good understanding of mathematical concepts like exponentiation and logarithms. The step-by-step process can help make solving them simpler:

• Rewrite the equation in a more manageable form using logarithms.
• Apply the change of base formula to make calculations easier.
• Calculate the logarithmic values using a calculator.
• Substitute and evaluate the expression to find \( x \).

In this particular example, calculating the logarithmic values helps find an approximate solution of \( x \approx -1.39 \), proving that solving with clarity and precision is possible even with complex equations.