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Scientific notation is a method used to write very large or very small numbers in a concise form. It expresses numbers as a product of two parts: a coefficient and a power of ten. This makes it easier to read and manage numbers, especially in fields like science and engineering.
For example, the number \(8.043 \times 10^{-8}\) consists of two parts: the coefficient \(8.043\) and the power of ten, \(10^{-8}\). The coefficient should be a number greater than or equal to 1 but less than 10. The power of ten indicates how many times you multiply the coefficient by ten. If the exponent is negative, like \(-8\) in this case, it means the coefficient is divided by ten multiple times. This helps to represent decimal numbers accurately.

• Example 1: \(123,000 = 1.23 \times 10^5\)
• Example 2: \(0.000076 = 7.6 \times 10^{-5}\)

Using scientific notation can simplify complex calculations and makes it straightforward to perform operations like multiplication or division with large numbers.

Logarithms are a way to understand the inverse of exponential growth. Common logarithms have a base of 10, often represented as \(\log\) without a subscript. They are crucial in simplifying multiplication into addition using their properties.
Here are some key properties:

• Product Rule: \(\log(a \times b) = \log(a) + \log(b)\). This property breaks down multiplication within a logarithm into the addition of two separate logs. It was used in the exercise to separate \(\log(8.043 \times 10^{-8})\) into \(\log(8.043) + \log(10^{-8})\).
• Power Rule: \(\log(a^n) = n \times \log(a)\). This helps when dealing with logarithms of powers.
• Quotient Rule: \(\log(\frac{a}{b}) = \log(a) - \log(b)\). This property simplifies division into subtraction within logarithms.

By utilizing these properties, complex problems can be simplified, making calculations more manageable.

When finding logarithms, a calculator can make the process quick and accurate. Most scientific calculators have a dedicated button for calculating common logarithms, usually labeled as "log". To find the logarithm of a number using a calculator, follow these steps:

• Step 1: Enter the number for which you want to find the logarithm. For instance, type \(8.043\).
• Step 2: Press the "log" button. The calculator will immediately provide the result, which in this case is \(\log(8.043) \approx 0.9056\).
• Step 3: For powers of ten, remember that \(\log(10^n) = n\). So for \(10^{-8}\), the calculation is straightforward: \(-8\).

By following these steps, you can accurately calculate logarithms and further apply the results to solve exercises such as finding \(\log(8.043 \times 10^{-8})\). Combining individual calculations using calculator results and log properties allows for efficient problem-solving.