91Ó°ÊÓ

In logarithms, the product rule makes it easier to break down complex products into simpler parts. This rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. More formally, for any positive numbers x and y and base b, the rule is expressed as:
ewline ewline \[ \log_b( xy ) = \log_b( x ) + \log_b( y ) \]

For instance, if you have \( \log_b (75) \) and you know the values of \( \log_b (3) \) and \( \log_b (5) \), you can simplify the computation process. First, note that 75 is the product of 3 and 25. So you can write:
ewline ewline \[ \log_b (75) = \log_b (3 \times 25) \]

Using the product rule of logarithms, you can split this into two simpler logarithms:
ewline ewline \[ \log_b (75) = \log_b (3) + \log_b (25) \]

This makes calculations much easier, especially when you know the individual logarithm values!

Another useful trick in logarithms is the power rule. This rule helps when you have a term raised to a power. According to the power rule, the logarithm of a number raised to an exponent is the exponent times the logarithm of the base number. Mathematically, it looks like this:
ewline ewline \[ \log_b (x^c) = c \cdot \log_b (x) \]

For example, consider \( \log_b (25) \). Since 25 can be written as \( 5^2 \), the problem becomes simpler:
ewline ewline \[ \log_b (25) = \log_b (5^2) \]

Now, apply the power rule to move the exponent to the front:
ewline ewline \[ \log_b (5^2) = 2 \cdot \log_b (5) \]

Therefore, knowing the value of \( \log_b (5) \) helps you easily find the value of \( \log_b (25) \) using this rule!

Combining the product and power rules of logarithms allows you to solve more complicated logarithmic expressions. Based on the exercise, you need to calculate \( \log_b (75) \). Here’s how you do it step by step:
ewline ewline

• First, recognize that 75 can be written as 3 times 25.
• Apply the product rule: \( \log_b (75) = \log_b (3 \times 25) = \log_b (3) + \log_b (25) \).
• Next, express 25 as a power: 25 is the same as \( 5^2 \).
• Apply the power rule to \( \log_b (25) \): \( \log_b (5^2) = 2 \cdot \log_b (5) \).
• Substitute the given values: \( \log_b (3) = 1.099 \) and \( \log_b (5) = 1.609 \).
• Perform the calculations: \( \log_b (25) = 2 \cdot 1.609 = 3.218 \).

Combining these, you get:
ewline ewline \[ \log_b (75) = 1.099 + 3.218 = 4.317 \]

By understanding these properties, you can handle various logarithmic problems with ease!